↑

Properties

Label 16-588e8-1.1-c5e8-0-1
Degree 1616
Conductor 1.429×10221.429\times 10^{22}
Sign 11
Analytic cond. 6.25609×10156.25609\times 10^{15}
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 36·3-s + 486·9-s − 462·11-s + 1.20e3·13-s − 228·17-s − 358·19-s − 2.14e3·23-s + 3.52e3·25-s − 1.10e4·29-s − 830·31-s + 1.66e4·33-s − 3.91e3·37-s − 4.33e4·39-s + 1.66e4·41-s − 2.90e4·43-s − 4.17e4·47-s + 8.20e3·51-s + 2.21e4·53-s + 1.28e4·57-s − 3.28e4·59-s − 8.37e4·61-s − 8.00e4·67-s + 7.73e4·69-s + 8.95e4·71-s + 2.24e4·73-s − 1.26e5·75-s − 7.52e4·79-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s − 1.15·11-s + 1.97·13-s − 0.191·17-s − 0.227·19-s − 0.846·23-s + 1.12·25-s − 2.44·29-s − 0.155·31-s + 2.65·33-s − 0.470·37-s − 4.56·39-s + 1.54·41-s − 2.39·43-s − 2.75·47-s + 0.441·51-s + 1.08·53-s + 0.525·57-s − 1.22·59-s − 2.88·61-s − 2.17·67-s + 1.95·69-s + 2.10·71-s + 0.493·73-s − 2.60·75-s − 1.35·79-s + ⋯

Functional equation

Λ(s)=((216⋅38⋅716)s/2 ΓC(s)8 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((216⋅38⋅716)s/2 ΓC(s+5/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 216⋅38⋅7162^{16} \cdot 3^{8} \cdot 7^{16}
Sign: 11
Analytic conductor: 6.25609×10156.25609\times 10^{15}
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅38⋅716, ( :[5/2]8), 1)(16,\ 2^{16} \cdot 3^{8} \cdot 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 2.2547161392.254716139
L(12)L(\frac12) ≈\approx 2.2547161392.254716139
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 (1+p2T+p4T2)4 ( 1 + p^{2} T + p^{4} T^{2} )^{4}
7 1 1
good5 1−3523T2−264p4T3−839647T4+877536p4T5+28930950702T6−1157319768p4T7−62249635398274T8−1157319768p9T9+28930950702p10T10+877536p19T11−839647p20T12−264p29T13−3523p30T14+p40T16 1 - 3523 T^{2} - 264 p^{4} T^{3} - 839647 T^{4} + 877536 p^{4} T^{5} + 28930950702 T^{6} - 1157319768 p^{4} T^{7} - 62249635398274 T^{8} - 1157319768 p^{9} T^{9} + 28930950702 p^{10} T^{10} + 877536 p^{19} T^{11} - 839647 p^{20} T^{12} - 264 p^{29} T^{13} - 3523 p^{30} T^{14} + p^{40} T^{16}
11 1+42pT−55261T2+23882490T3+30394966829T4+4313182607736T5+7711402699001610T6+1745859627280956732T7− 1 + 42 p T - 55261 T^{2} + 23882490 T^{3} + 30394966829 T^{4} + 4313182607736 T^{5} + 7711402699001610 T^{6} + 1745859627280956732 T^{7} - 81 ⁣⋯ ⁣6281\!\cdots\!62T8+1745859627280956732p5T9+7711402699001610p10T10+4313182607736p15T11+30394966829p20T12+23882490p25T13−55261p30T14+42p36T15+p40T16 T^{8} + 1745859627280956732 p^{5} T^{9} + 7711402699001610 p^{10} T^{10} + 4313182607736 p^{15} T^{11} + 30394966829 p^{20} T^{12} + 23882490 p^{25} T^{13} - 55261 p^{30} T^{14} + 42 p^{36} T^{15} + p^{40} T^{16}
13 (1−602T−12679pT2−145984794T3+357680241104T4−145984794p5T5−12679p11T6−602p15T7+p20T8)2 ( 1 - 602 T - 12679 p T^{2} - 145984794 T^{3} + 357680241104 T^{4} - 145984794 p^{5} T^{5} - 12679 p^{11} T^{6} - 602 p^{15} T^{7} + p^{20} T^{8} )^{2}
17 1+228T−3817300T2−1812530376T3+7723728660698T4+3961559277827628T5−9874383121499354160T6− 1 + 228 T - 3817300 T^{2} - 1812530376 T^{3} + 7723728660698 T^{4} + 3961559277827628 T^{5} - 9874383121499354160 T^{6} - 30 ⁣⋯ ⁣8830\!\cdots\!88T7+ T^{7} + 11 ⁣⋯ ⁣6311\!\cdots\!63T8− T^{8} - 30 ⁣⋯ ⁣8830\!\cdots\!88p5T9−9874383121499354160p10T10+3961559277827628p15T11+7723728660698p20T12−1812530376p25T13−3817300p30T14+228p35T15+p40T16 p^{5} T^{9} - 9874383121499354160 p^{10} T^{10} + 3961559277827628 p^{15} T^{11} + 7723728660698 p^{20} T^{12} - 1812530376 p^{25} T^{13} - 3817300 p^{30} T^{14} + 228 p^{35} T^{15} + p^{40} T^{16}
19 1+358T−7237113T2−2935172086T3+28171845756353T4+9461877710293488T5−89100991373462587298T6− 1 + 358 T - 7237113 T^{2} - 2935172086 T^{3} + 28171845756353 T^{4} + 9461877710293488 T^{5} - 89100991373462587298 T^{6} - 10 ⁣⋯ ⁣2010\!\cdots\!20T7+ T^{7} + 24 ⁣⋯ ⁣4224\!\cdots\!42T8− T^{8} - 10 ⁣⋯ ⁣2010\!\cdots\!20p5T9−89100991373462587298p10T10+9461877710293488p15T11+28171845756353p20T12−2935172086p25T13−7237113p30T14+358p35T15+p40T16 p^{5} T^{9} - 89100991373462587298 p^{10} T^{10} + 9461877710293488 p^{15} T^{11} + 28171845756353 p^{20} T^{12} - 2935172086 p^{25} T^{13} - 7237113 p^{30} T^{14} + 358 p^{35} T^{15} + p^{40} T^{16}
23 1+2148T−12238924T2+4077683208T3+128948594947802T4−143919044174116500T5−26075111207770425456pT6+ 1 + 2148 T - 12238924 T^{2} + 4077683208 T^{3} + 128948594947802 T^{4} - 143919044174116500 T^{5} - 26075111207770425456 p T^{6} + 52 ⁣⋯ ⁣7252\!\cdots\!72T7+ T^{7} + 18 ⁣⋯ ⁣3118\!\cdots\!31T8+ T^{8} + 52 ⁣⋯ ⁣7252\!\cdots\!72p5T9−26075111207770425456p11T10−143919044174116500p15T11+128948594947802p20T12+4077683208p25T13−12238924p30T14+2148p35T15+p40T16 p^{5} T^{9} - 26075111207770425456 p^{11} T^{10} - 143919044174116500 p^{15} T^{11} + 128948594947802 p^{20} T^{12} + 4077683208 p^{25} T^{13} - 12238924 p^{30} T^{14} + 2148 p^{35} T^{15} + p^{40} T^{16}
29 (1+5532T+56710687T2+131401968540T3+1151610033241988T4+131401968540p5T5+56710687p10T6+5532p15T7+p20T8)2 ( 1 + 5532 T + 56710687 T^{2} + 131401968540 T^{3} + 1151610033241988 T^{4} + 131401968540 p^{5} T^{5} + 56710687 p^{10} T^{6} + 5532 p^{15} T^{7} + p^{20} T^{8} )^{2}
31 1+830T−30566692T2+460416975904T3+906611510861135T4−13425215480714328836T5+ 1 + 830 T - 30566692 T^{2} + 460416975904 T^{3} + 906611510861135 T^{4} - 13425215480714328836 T^{5} + 95 ⁣⋯ ⁣4095\!\cdots\!40T6+ T^{6} + 37 ⁣⋯ ⁣3037\!\cdots\!30T7− T^{7} - 27 ⁣⋯ ⁣2827\!\cdots\!28T8+ T^{8} + 37 ⁣⋯ ⁣3037\!\cdots\!30p5T9+ p^{5} T^{9} + 95 ⁣⋯ ⁣4095\!\cdots\!40p10T10−13425215480714328836p15T11+906611510861135p20T12+460416975904p25T13−30566692p30T14+830p35T15+p40T16 p^{10} T^{10} - 13425215480714328836 p^{15} T^{11} + 906611510861135 p^{20} T^{12} + 460416975904 p^{25} T^{13} - 30566692 p^{30} T^{14} + 830 p^{35} T^{15} + p^{40} T^{16}
37 1+3914T−1798245pT2−1097998070882T3−1825206898694683T4+62220290884192688868T5+ 1 + 3914 T - 1798245 p T^{2} - 1097998070882 T^{3} - 1825206898694683 T^{4} + 62220290884192688868 T^{5} + 62 ⁣⋯ ⁣6662\!\cdots\!66T6− T^{6} - 11 ⁣⋯ ⁣8811\!\cdots\!88T7− T^{7} - 46 ⁣⋯ ⁣3846\!\cdots\!38T8− T^{8} - 11 ⁣⋯ ⁣8811\!\cdots\!88p5T9+ p^{5} T^{9} + 62 ⁣⋯ ⁣6662\!\cdots\!66p10T10+62220290884192688868p15T11−1825206898694683p20T12−1097998070882p25T13−1798245p31T14+3914p35T15+p40T16 p^{10} T^{10} + 62220290884192688868 p^{15} T^{11} - 1825206898694683 p^{20} T^{12} - 1097998070882 p^{25} T^{13} - 1798245 p^{31} T^{14} + 3914 p^{35} T^{15} + p^{40} T^{16}
41 (1−8316T+131719112T2+1793338464780T3−11507939827736466T4+1793338464780p5T5+131719112p10T6−8316p15T7+p20T8)2 ( 1 - 8316 T + 131719112 T^{2} + 1793338464780 T^{3} - 11507939827736466 T^{4} + 1793338464780 p^{5} T^{5} + 131719112 p^{10} T^{6} - 8316 p^{15} T^{7} + p^{20} T^{8} )^{2}
43 (1+14518T+553805833T2+5594375858722T3+119245786138973608T4+5594375858722p5T5+553805833p10T6+14518p15T7+p20T8)2 ( 1 + 14518 T + 553805833 T^{2} + 5594375858722 T^{3} + 119245786138973608 T^{4} + 5594375858722 p^{5} T^{5} + 553805833 p^{10} T^{6} + 14518 p^{15} T^{7} + p^{20} T^{8} )^{2}
47 1+41700T+537485524T2+6358172548200T3+162169996689714634T4+ 1 + 41700 T + 537485524 T^{2} + 6358172548200 T^{3} + 162169996689714634 T^{4} + 49 ⁣⋯ ⁣0049\!\cdots\!00T5− T^{5} - 48 ⁣⋯ ⁣4448\!\cdots\!44T6− T^{6} - 17 ⁣⋯ ⁣0017\!\cdots\!00pT7− p T^{7} - 10 ⁣⋯ ⁣6110\!\cdots\!61T8− T^{8} - 17 ⁣⋯ ⁣0017\!\cdots\!00p6T9− p^{6} T^{9} - 48 ⁣⋯ ⁣4448\!\cdots\!44p10T10+ p^{10} T^{10} + 49 ⁣⋯ ⁣0049\!\cdots\!00p15T11+162169996689714634p20T12+6358172548200p25T13+537485524p30T14+41700p35T15+p40T16 p^{15} T^{11} + 162169996689714634 p^{20} T^{12} + 6358172548200 p^{25} T^{13} + 537485524 p^{30} T^{14} + 41700 p^{35} T^{15} + p^{40} T^{16}
53 1−22164T−121257815T2−31884451303188T3+713358687755937853T4+ 1 - 22164 T - 121257815 T^{2} - 31884451303188 T^{3} + 713358687755937853 T^{4} + 34 ⁣⋯ ⁣3634\!\cdots\!36T5+ T^{5} + 46 ⁣⋯ ⁣1046\!\cdots\!10T6− T^{6} - 99 ⁣⋯ ⁣2499\!\cdots\!24T7− T^{7} - 41 ⁣⋯ ⁣0241\!\cdots\!02T8− T^{8} - 99 ⁣⋯ ⁣2499\!\cdots\!24p5T9+ p^{5} T^{9} + 46 ⁣⋯ ⁣1046\!\cdots\!10p10T10+ p^{10} T^{10} + 34 ⁣⋯ ⁣3634\!\cdots\!36p15T11+713358687755937853p20T12−31884451303188p25T13−121257815p30T14−22164p35T15+p40T16 p^{15} T^{11} + 713358687755937853 p^{20} T^{12} - 31884451303188 p^{25} T^{13} - 121257815 p^{30} T^{14} - 22164 p^{35} T^{15} + p^{40} T^{16}
59 1+32886T−1083783697T2−35352420196734T3+734569909452509033T4+ 1 + 32886 T - 1083783697 T^{2} - 35352420196734 T^{3} + 734569909452509033 T^{4} + 86 ⁣⋯ ⁣9286\!\cdots\!92T5− T^{5} - 96 ⁣⋯ ⁣0296\!\cdots\!02T6+ T^{6} + 76 ⁣⋯ ⁣2876\!\cdots\!28T7+ T^{7} + 95 ⁣⋯ ⁣1495\!\cdots\!14T8+ T^{8} + 76 ⁣⋯ ⁣2876\!\cdots\!28p5T9− p^{5} T^{9} - 96 ⁣⋯ ⁣0296\!\cdots\!02p10T10+ p^{10} T^{10} + 86 ⁣⋯ ⁣9286\!\cdots\!92p15T11+734569909452509033p20T12−35352420196734p25T13−1083783697p30T14+32886p35T15+p40T16 p^{15} T^{11} + 734569909452509033 p^{20} T^{12} - 35352420196734 p^{25} T^{13} - 1083783697 p^{30} T^{14} + 32886 p^{35} T^{15} + p^{40} T^{16}
61 1+83732T+1872200780T2+2539765176280T3+1308252811325909786T4+ 1 + 83732 T + 1872200780 T^{2} + 2539765176280 T^{3} + 1308252811325909786 T^{4} + 75 ⁣⋯ ⁣5275\!\cdots\!52T5+ T^{5} + 99 ⁣⋯ ⁣4899\!\cdots\!48T6+ T^{6} + 38 ⁣⋯ ⁣3638\!\cdots\!36T7+ T^{7} + 21 ⁣⋯ ⁣1921\!\cdots\!19T8+ T^{8} + 38 ⁣⋯ ⁣3638\!\cdots\!36p5T9+ p^{5} T^{9} + 99 ⁣⋯ ⁣4899\!\cdots\!48p10T10+ p^{10} T^{10} + 75 ⁣⋯ ⁣5275\!\cdots\!52p15T11+1308252811325909786p20T12+2539765176280p25T13+1872200780p30T14+83732p35T15+p40T16 p^{15} T^{11} + 1308252811325909786 p^{20} T^{12} + 2539765176280 p^{25} T^{13} + 1872200780 p^{30} T^{14} + 83732 p^{35} T^{15} + p^{40} T^{16}
67 1+80034T+295144795T2−67594226392002T3+2776430208831340477T4+ 1 + 80034 T + 295144795 T^{2} - 67594226392002 T^{3} + 2776430208831340477 T^{4} + 10 ⁣⋯ ⁣4810\!\cdots\!48T5− T^{5} - 54 ⁣⋯ ⁣9454\!\cdots\!94T6− T^{6} - 84 ⁣⋯ ⁣8884\!\cdots\!88T7+ T^{7} + 64 ⁣⋯ ⁣8264\!\cdots\!82T8− T^{8} - 84 ⁣⋯ ⁣8884\!\cdots\!88p5T9− p^{5} T^{9} - 54 ⁣⋯ ⁣9454\!\cdots\!94p10T10+ p^{10} T^{10} + 10 ⁣⋯ ⁣4810\!\cdots\!48p15T11+2776430208831340477p20T12−67594226392002p25T13+295144795p30T14+80034p35T15+p40T16 p^{15} T^{11} + 2776430208831340477 p^{20} T^{12} - 67594226392002 p^{25} T^{13} + 295144795 p^{30} T^{14} + 80034 p^{35} T^{15} + p^{40} T^{16}
71 (1−44772T+5603702956T2−219545846683812T3+14130153190557486902T4−219545846683812p5T5+5603702956p10T6−44772p15T7+p20T8)2 ( 1 - 44772 T + 5603702956 T^{2} - 219545846683812 T^{3} + 14130153190557486902 T^{4} - 219545846683812 p^{5} T^{5} + 5603702956 p^{10} T^{6} - 44772 p^{15} T^{7} + p^{20} T^{8} )^{2}
73 1−22470T−3227884709T2−55184471402730T3+5281219798633591705T4+ 1 - 22470 T - 3227884709 T^{2} - 55184471402730 T^{3} + 5281219798633591705 T^{4} + 29 ⁣⋯ ⁣4029\!\cdots\!40T5+ T^{5} + 52 ⁣⋯ ⁣9852\!\cdots\!98T6− T^{6} - 45 ⁣⋯ ⁣6045\!\cdots\!60T7− T^{7} - 39 ⁣⋯ ⁣9439\!\cdots\!94T8− T^{8} - 45 ⁣⋯ ⁣6045\!\cdots\!60p5T9+ p^{5} T^{9} + 52 ⁣⋯ ⁣9852\!\cdots\!98p10T10+ p^{10} T^{10} + 29 ⁣⋯ ⁣4029\!\cdots\!40p15T11+5281219798633591705p20T12−55184471402730p25T13−3227884709p30T14−22470p35T15+p40T16 p^{15} T^{11} + 5281219798633591705 p^{20} T^{12} - 55184471402730 p^{25} T^{13} - 3227884709 p^{30} T^{14} - 22470 p^{35} T^{15} + p^{40} T^{16}
79 1+75286T−4589904580T2−271785361643200T3+23588797479279670751T4+ 1 + 75286 T - 4589904580 T^{2} - 271785361643200 T^{3} + 23588797479279670751 T^{4} + 58 ⁣⋯ ⁣5658\!\cdots\!56T5− T^{5} - 88 ⁣⋯ ⁣2888\!\cdots\!28T6− T^{6} - 13 ⁣⋯ ⁣7813\!\cdots\!78T7+ T^{7} + 19 ⁣⋯ ⁣8419\!\cdots\!84T8− T^{8} - 13 ⁣⋯ ⁣7813\!\cdots\!78p5T9− p^{5} T^{9} - 88 ⁣⋯ ⁣2888\!\cdots\!28p10T10+ p^{10} T^{10} + 58 ⁣⋯ ⁣5658\!\cdots\!56p15T11+23588797479279670751p20T12−271785361643200p25T13−4589904580p30T14+75286p35T15+p40T16 p^{15} T^{11} + 23588797479279670751 p^{20} T^{12} - 271785361643200 p^{25} T^{13} - 4589904580 p^{30} T^{14} + 75286 p^{35} T^{15} + p^{40} T^{16}
83 (1−17418T+11662792993T2−252243184391142T3+61381737302005449608T4−252243184391142p5T5+11662792993p10T6−17418p15T7+p20T8)2 ( 1 - 17418 T + 11662792993 T^{2} - 252243184391142 T^{3} + 61381737302005449608 T^{4} - 252243184391142 p^{5} T^{5} + 11662792993 p^{10} T^{6} - 17418 p^{15} T^{7} + p^{20} T^{8} )^{2}
89 1+28944T−4147498688T2−127105851999456T3+27498803712269304034T4+ 1 + 28944 T - 4147498688 T^{2} - 127105851999456 T^{3} + 27498803712269304034 T^{4} + 18 ⁣⋯ ⁣2418\!\cdots\!24T5+ T^{5} + 34 ⁣⋯ ⁣7634\!\cdots\!76T6+ T^{6} + 10 ⁣⋯ ⁣6810\!\cdots\!68T7− T^{7} - 15 ⁣⋯ ⁣2915\!\cdots\!29T8+ T^{8} + 10 ⁣⋯ ⁣6810\!\cdots\!68p5T9+ p^{5} T^{9} + 34 ⁣⋯ ⁣7634\!\cdots\!76p10T10+ p^{10} T^{10} + 18 ⁣⋯ ⁣2418\!\cdots\!24p15T11+27498803712269304034p20T12−127105851999456p25T13−4147498688p30T14+28944p35T15+p40T16 p^{15} T^{11} + 27498803712269304034 p^{20} T^{12} - 127105851999456 p^{25} T^{13} - 4147498688 p^{30} T^{14} + 28944 p^{35} T^{15} + p^{40} T^{16}
97 (1−216678T+43602999305T2−4974916579453302T3+ ( 1 - 216678 T + 43602999305 T^{2} - 4974916579453302 T^{3} + 56 ⁣⋯ ⁣0056\!\cdots\!00T4−4974916579453302p5T5+43602999305p10T6−216678p15T7+p20T8)2 T^{4} - 4974916579453302 p^{5} T^{5} + 43602999305 p^{10} T^{6} - 216678 p^{15} T^{7} + p^{20} T^{8} )^{2}
show more
show less
   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.78725611736600871477000197084, −3.40804613057975487160854921369, −3.39228616223431254588499237447, −3.38470857199495160400649404749, −3.38358124510480296654621372773, −3.22921224745443967011301324273, −3.11286854645197789354337787934, −3.09511539833759666465238796610, −2.68181713973450480556465538386, −2.30236535280910748664620517995, −2.27216269510107732894365325685, −2.25282808466384982199113150622, −1.93353275365786013877437808135, −1.78406034225996911245922485760, −1.75796981814511933371357216513, −1.66493122688237954049324871011, −1.46324361554790387047470707127, −1.26700399564779224186915229426, −0.937635321912667336934340148633, −0.76584839327575808548131041259, −0.70568049398851539419256735121, −0.51533397506853286279368896396, −0.44926658143144563230536709623, −0.26767343996333052369294636123, −0.16350537271178732751894324320, 0.16350537271178732751894324320, 0.26767343996333052369294636123, 0.44926658143144563230536709623, 0.51533397506853286279368896396, 0.70568049398851539419256735121, 0.76584839327575808548131041259, 0.937635321912667336934340148633, 1.26700399564779224186915229426, 1.46324361554790387047470707127, 1.66493122688237954049324871011, 1.75796981814511933371357216513, 1.78406034225996911245922485760, 1.93353275365786013877437808135, 2.25282808466384982199113150622, 2.27216269510107732894365325685, 2.30236535280910748664620517995, 2.68181713973450480556465538386, 3.09511539833759666465238796610, 3.11286854645197789354337787934, 3.22921224745443967011301324273, 3.38358124510480296654621372773, 3.38470857199495160400649404749, 3.39228616223431254588499237447, 3.40804613057975487160854921369, 3.78725611736600871477000197084

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.