| L(s) = 1 | + 2·2-s + 2·4-s − 22·7-s + 4·8-s + 15·9-s − 6·11-s − 12·13-s − 44·14-s + 8·16-s + 78·17-s + 30·18-s − 58·19-s − 12·22-s + 12·23-s − 24·26-s − 44·28-s − 54·29-s − 128·31-s + 8·32-s + 156·34-s + 30·36-s + 40·37-s − 116·38-s + 120·43-s − 12·44-s + 24·46-s + 132·47-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 3.14·7-s + 1/2·8-s + 5/3·9-s − 0.545·11-s − 0.923·13-s − 3.14·14-s + 1/2·16-s + 4.58·17-s + 5/3·18-s − 3.05·19-s − 0.545·22-s + 0.521·23-s − 0.923·26-s − 1.57·28-s − 1.86·29-s − 4.12·31-s + 1/4·32-s + 4.58·34-s + 5/6·36-s + 1.08·37-s − 3.05·38-s + 2.79·43-s − 0.272·44-s + 0.521·46-s + 2.80·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.047135233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047135233\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 12 T + 14 p T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 5 p T^{2} + 16 p^{2} T^{4} - 5 p^{5} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 686 T^{4} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 22 T + 221 T^{2} + 1362 T^{3} + 8024 T^{4} + 1362 p^{2} T^{5} + 221 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} - 1710 T^{3} - 16024 T^{4} - 1710 p^{2} T^{5} + 45 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 78 T + 181 p T^{2} - 81822 T^{3} + 1602972 T^{4} - 81822 p^{2} T^{5} + 181 p^{5} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 58 T + 1325 T^{2} + 12942 T^{3} + 74504 T^{4} + 12942 p^{2} T^{5} + 1325 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 677 T^{2} - 7548 T^{3} + 141192 T^{4} - 7548 p^{2} T^{5} + 677 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 54 T + 1273 T^{2} - 2106 T^{3} - 460188 T^{4} - 2106 p^{2} T^{5} + 1273 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 128 T + 8192 T^{2} + 365952 T^{3} + 12745358 T^{4} + 365952 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40 T + 401 T^{2} + 52236 T^{3} - 3248140 T^{4} + 52236 p^{2} T^{5} + 401 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 1521 T^{2} + 91572 T^{3} + 840356 T^{4} + 91572 p^{2} T^{5} + 1521 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 120 T + 8177 T^{2} - 405240 T^{3} + 16860528 T^{4} - 405240 p^{2} T^{5} + 8177 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 84 T + 5930 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 1305 T^{2} - 208794 T^{3} - 5098528 T^{4} - 208794 p^{2} T^{5} + 1305 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 78 T - 2771 T^{2} - 110214 T^{3} + 41926620 T^{4} - 110214 p^{2} T^{5} - 2771 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 86 T + 4985 T^{2} - 290202 T^{3} + 7709888 T^{4} - 290202 p^{2} T^{5} + 4985 p^{4} T^{6} - 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 42 T + 3357 T^{2} - 375150 T^{3} + 3657944 T^{4} - 375150 p^{2} T^{5} + 3357 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 136 T + 9248 T^{2} + 444312 T^{3} + 17094734 T^{4} + 444312 p^{2} T^{5} + 9248 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 96 T + 10898 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 192 T + 18432 T^{2} - 1598016 T^{3} + 136488302 T^{4} - 1598016 p^{2} T^{5} + 18432 p^{4} T^{6} - 192 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 60 T + 5661 T^{2} - 927072 T^{3} - 53599444 T^{4} - 927072 p^{2} T^{5} + 5661 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 280 T + 24089 T^{2} + 773796 T^{3} - 268596364 T^{4} + 773796 p^{2} T^{5} + 24089 p^{4} T^{6} - 280 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.89145518612938970727692159288, −12.66271162178600976518060930701, −12.43893292379960252889062441682, −12.40595888421063200294191335786, −12.30987150271449776107319815442, −11.18888117674074169664036326344, −10.76596277264235722033582706737, −10.45145477118621779010823865466, −10.43282061106278180731677964546, −9.681118521230812636999806410897, −9.641565124467089165049291160972, −9.238107072771251347498232119849, −9.129428276412286806304527746000, −7.80562933100406125597898926348, −7.69368741356450004548514797437, −7.36696718885324987891037216057, −7.07571546754710750366822790690, −6.31424320090316127341430914194, −6.07665860600058424037499978925, −5.46016289798215233036269888470, −5.31972203220906167945160620566, −4.07335077191326694410777982992, −3.72142540296874082628417979044, −3.53781147181532421255962996871, −2.44872389725305077785373347331,
2.44872389725305077785373347331, 3.53781147181532421255962996871, 3.72142540296874082628417979044, 4.07335077191326694410777982992, 5.31972203220906167945160620566, 5.46016289798215233036269888470, 6.07665860600058424037499978925, 6.31424320090316127341430914194, 7.07571546754710750366822790690, 7.36696718885324987891037216057, 7.69368741356450004548514797437, 7.80562933100406125597898926348, 9.129428276412286806304527746000, 9.238107072771251347498232119849, 9.641565124467089165049291160972, 9.681118521230812636999806410897, 10.43282061106278180731677964546, 10.45145477118621779010823865466, 10.76596277264235722033582706737, 11.18888117674074169664036326344, 12.30987150271449776107319815442, 12.40595888421063200294191335786, 12.43893292379960252889062441682, 12.66271162178600976518060930701, 12.89145518612938970727692159288
