| L(s) = 1 | − 3.12e4·5-s + 2.28e5·7-s − 3.39e6·11-s + 9.73e6·13-s − 4.88e7·17-s + 9.26e7·19-s − 3.24e8·23-s + 7.32e8·25-s − 1.19e9·29-s + 1.69e9·31-s − 7.13e9·35-s + 3.17e9·37-s − 5.41e9·41-s + 2.69e9·43-s + 1.10e10·47-s − 7.36e10·49-s + 7.43e10·53-s + 1.06e11·55-s + 1.11e11·59-s + 3.89e11·61-s − 3.04e11·65-s + 1.36e12·67-s − 1.52e12·71-s + 2.52e12·73-s − 7.75e11·77-s + 3.35e12·79-s − 6.16e12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.733·7-s − 0.577·11-s + 0.559·13-s − 0.491·17-s + 0.451·19-s − 0.457·23-s + 3/5·25-s − 0.374·29-s + 0.342·31-s − 0.656·35-s + 0.203·37-s − 0.177·41-s + 0.0649·43-s + 0.149·47-s − 0.760·49-s + 0.460·53-s + 0.516·55-s + 0.344·59-s + 0.966·61-s − 0.500·65-s + 1.84·67-s − 1.41·71-s + 1.95·73-s − 0.423·77-s + 1.55·79-s − 2.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 228472 T + 17982545730 p T^{2} - 228472 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 308472 p T + 270611827318 p^{2} T^{2} + 308472 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 748660 p T + 619636033756206 T^{2} - 748660 p^{14} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48892572 T + 19975394071818070 T^{2} + 48892572 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 92678560 T + 81153360788168118 T^{2} - 92678560 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 324639912 T + 113441166751669102 T^{2} + 324639912 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1198137468 T + 16080611455697567134 T^{2} + 1198137468 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1691547208 T + 42948222438766145598 T^{2} - 1691547208 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3176153716 T + \)\(48\!\cdots\!58\)\( T^{2} - 3176153716 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 131974500 p T + \)\(18\!\cdots\!42\)\( T^{2} + 131974500 p^{14} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2693787832 T - \)\(10\!\cdots\!58\)\( T^{2} - 2693787832 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11041050072 T + \)\(64\!\cdots\!50\)\( T^{2} - 11041050072 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 74345936196 T + \)\(53\!\cdots\!50\)\( T^{2} - 74345936196 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 111691086552 T + \)\(50\!\cdots\!34\)\( T^{2} - 111691086552 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 389105330764 T + \)\(36\!\cdots\!86\)\( T^{2} - 389105330764 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1365934012168 T + \)\(14\!\cdots\!30\)\( T^{2} - 1365934012168 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1522323635280 T + \)\(28\!\cdots\!22\)\( T^{2} + 1522323635280 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2523552191716 T + \)\(47\!\cdots\!30\)\( T^{2} - 2523552191716 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3351472043416 T + \)\(11\!\cdots\!42\)\( T^{2} - 3351472043416 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6162816362952 T + \)\(24\!\cdots\!02\)\( T^{2} + 6162816362952 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6249390546180 T + \)\(49\!\cdots\!38\)\( T^{2} + 6249390546180 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1103963471524 T + \)\(82\!\cdots\!98\)\( T^{2} - 1103963471524 p^{13} T^{3} + p^{26} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.911196071207612506780026965862, −9.785979087385831864286589497425, −8.831476825810461247469609652571, −8.577241621159751779921066492091, −7.892780290773090018578042160645, −7.86748807660932256416418938193, −7.03271323530149924256854702073, −6.69240296978012039872515083853, −5.92718434745668335132298875008, −5.39565687622311328121744164783, −4.85503426826326578983988765058, −4.41458658578043256522327605725, −3.65450785022310253666699267313, −3.52149280211512392281226396265, −2.41274856419441983060874383219, −2.33189415013212057325632674640, −1.20643040365307642217499870014, −1.10229806877567368693622519018, 0, 0,
1.10229806877567368693622519018, 1.20643040365307642217499870014, 2.33189415013212057325632674640, 2.41274856419441983060874383219, 3.52149280211512392281226396265, 3.65450785022310253666699267313, 4.41458658578043256522327605725, 4.85503426826326578983988765058, 5.39565687622311328121744164783, 5.92718434745668335132298875008, 6.69240296978012039872515083853, 7.03271323530149924256854702073, 7.86748807660932256416418938193, 7.892780290773090018578042160645, 8.577241621159751779921066492091, 8.831476825810461247469609652571, 9.785979087385831864286589497425, 9.911196071207612506780026965862
