| L(s) = 1 | − 16·4-s + 44·9-s + 42·11-s + 160·16-s − 346·19-s + 236·29-s + 34·31-s − 704·36-s + 278·41-s − 672·44-s + 1.49e3·49-s + 906·59-s − 654·61-s − 1.28e3·64-s + 390·71-s + 5.53e3·76-s + 2.28e3·79-s + 832·81-s + 4.34e3·89-s + 1.84e3·99-s + 2.19e3·101-s + 4.55e3·109-s − 3.77e3·116-s − 4.11e3·121-s − 544·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.62·9-s + 1.15·11-s + 5/2·16-s − 4.17·19-s + 1.51·29-s + 0.196·31-s − 3.25·36-s + 1.05·41-s − 2.30·44-s + 4.36·49-s + 1.99·59-s − 1.37·61-s − 5/2·64-s + 0.651·71-s + 8.35·76-s + 3.24·79-s + 1.14·81-s + 5.16·89-s + 1.87·99-s + 2.15·101-s + 3.99·109-s − 3.02·116-s − 3.09·121-s − 0.393·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.600376373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600376373\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 3 | \( 1 - 44 T^{2} + 368 p T^{4} + 9445 T^{6} - 633224 T^{8} + 9445 p^{6} T^{10} + 368 p^{13} T^{12} - 44 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 - 1496 T^{2} + 1135864 T^{4} - 1770145 p^{3} T^{6} + 100951076 p^{4} T^{8} - 1770145 p^{9} T^{10} + 1135864 p^{12} T^{12} - 1496 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 - 21 T + 2719 T^{2} - 126987 T^{3} + 3530624 T^{4} - 126987 p^{3} T^{5} + 2719 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 13248 T^{2} + 79685888 T^{4} - 294284970279 T^{6} + 758213803513032 T^{8} - 294284970279 p^{6} T^{10} + 79685888 p^{12} T^{12} - 13248 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 19104 T^{2} + 11693324 p T^{4} - 1447730827543 T^{6} + 8058245392097268 T^{8} - 1447730827543 p^{6} T^{10} + 11693324 p^{13} T^{12} - 19104 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 + 173 T + 25377 T^{2} + 2932999 T^{3} + 264013708 T^{4} + 2932999 p^{3} T^{5} + 25377 p^{6} T^{6} + 173 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 118 T + 61751 T^{2} - 8515068 T^{3} + 1867055088 T^{4} - 8515068 p^{3} T^{5} + 61751 p^{6} T^{6} - 118 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 17 T + 11172 T^{2} - 1640140 T^{3} + 1568834653 T^{4} - 1640140 p^{3} T^{5} + 11172 p^{6} T^{6} - 17 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 87647 T^{2} + 3721427950 T^{4} - 143410729258201 T^{6} + 9437452931486063618 T^{8} - 143410729258201 p^{6} T^{10} + 3721427950 p^{12} T^{12} - 87647 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 - 139 T + 118058 T^{2} - 18482874 T^{3} + 11562193671 T^{4} - 18482874 p^{3} T^{5} + 118058 p^{6} T^{6} - 139 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 218884 T^{2} + 33171161428 T^{4} - 3698987179964380 T^{6} + \)\(33\!\cdots\!62\)\( T^{8} - 3698987179964380 p^{6} T^{10} + 33171161428 p^{12} T^{12} - 218884 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 419427 T^{2} + 71519170966 T^{4} - 6679106473246381 T^{6} + \)\(55\!\cdots\!34\)\( T^{8} - 6679106473246381 p^{6} T^{10} + 71519170966 p^{12} T^{12} - 419427 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 842163 T^{2} + 346805145106 T^{4} - 90053992303419781 T^{6} + \)\(16\!\cdots\!66\)\( T^{8} - 90053992303419781 p^{6} T^{10} + 346805145106 p^{12} T^{12} - 842163 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 - 453 T + 512870 T^{2} - 262193049 T^{3} + 135268242162 T^{4} - 262193049 p^{3} T^{5} + 512870 p^{6} T^{6} - 453 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 327 T + 703169 T^{2} + 181008321 T^{3} + 214272589464 T^{4} + 181008321 p^{3} T^{5} + 703169 p^{6} T^{6} + 327 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 1687255 T^{2} + 1267503561658 T^{4} - 586088665406247409 T^{6} + \)\(19\!\cdots\!02\)\( T^{8} - 586088665406247409 p^{6} T^{10} + 1267503561658 p^{12} T^{12} - 1687255 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 195 T + 725314 T^{2} - 106317378 T^{3} + 5198032027 p T^{4} - 106317378 p^{3} T^{5} + 725314 p^{6} T^{6} - 195 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 2257679 T^{2} + 2455843413142 T^{4} - 1672355827875771601 T^{6} + \)\(77\!\cdots\!26\)\( T^{8} - 1672355827875771601 p^{6} T^{10} + 2455843413142 p^{12} T^{12} - 2257679 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 1140 T + 1445260 T^{2} - 1359377572 T^{3} + 999599980326 T^{4} - 1359377572 p^{3} T^{5} + 1445260 p^{6} T^{6} - 1140 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 3572711 T^{2} + 5744228468162 T^{4} - 5616336920489921745 T^{6} + \)\(37\!\cdots\!94\)\( T^{8} - 5616336920489921745 p^{6} T^{10} + 5744228468162 p^{12} T^{12} - 3572711 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 2170 T + 3806124 T^{2} - 4376259102 T^{3} + 4366516459126 T^{4} - 4376259102 p^{3} T^{5} + 3806124 p^{6} T^{6} - 2170 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 5390921 T^{2} + 13734119381419 T^{4} - 21832765071125953075 T^{6} + \)\(23\!\cdots\!36\)\( T^{8} - 21832765071125953075 p^{6} T^{10} + 13734119381419 p^{12} T^{12} - 5390921 p^{18} T^{14} + p^{24} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.96344155895369380810480016867, −3.75994373949286782253252079362, −3.66307200059806434440105768441, −3.28298888543284357786682541665, −3.24643849852143620491621538596, −3.16189881202455403661142440593, −3.09295824816688349269955779633, −3.00911281892576539223792016648, −2.60689671665312358013737664324, −2.43349055533756103538051719865, −2.32341866345266707873891956109, −2.21916743457485442089499809077, −1.92895323824161830021227030763, −1.89278883556974537204132910284, −1.84078277377538880612354801428, −1.82830380867609371353478329579, −1.73025246738755659651381313189, −1.05808425363997772378796888485, −0.882113993166037752937518040873, −0.878206225962369372861895647440, −0.819325275377825849433317100462, −0.71844524953528603692237871734, −0.57611388679081061203647963712, −0.48062945996173293681499687813, −0.05613313390177717212234510882,
0.05613313390177717212234510882, 0.48062945996173293681499687813, 0.57611388679081061203647963712, 0.71844524953528603692237871734, 0.819325275377825849433317100462, 0.878206225962369372861895647440, 0.882113993166037752937518040873, 1.05808425363997772378796888485, 1.73025246738755659651381313189, 1.82830380867609371353478329579, 1.84078277377538880612354801428, 1.89278883556974537204132910284, 1.92895323824161830021227030763, 2.21916743457485442089499809077, 2.32341866345266707873891956109, 2.43349055533756103538051719865, 2.60689671665312358013737664324, 3.00911281892576539223792016648, 3.09295824816688349269955779633, 3.16189881202455403661142440593, 3.24643849852143620491621538596, 3.28298888543284357786682541665, 3.66307200059806434440105768441, 3.75994373949286782253252079362, 3.96344155895369380810480016867
Plot not available for L-functions of degree greater than 10.
