| L(s) = 1 | + 8·2-s + 210·5-s − 1.01e3·7-s − 512·8-s + 1.68e3·10-s − 1.09e3·11-s − 1.38e3·13-s − 8.12e3·14-s − 4.09e3·16-s + 2.94e4·17-s − 7.98e4·19-s − 8.73e3·22-s − 6.87e4·23-s + 7.81e4·25-s − 1.10e4·26-s + 1.02e5·29-s − 2.27e5·31-s + 2.35e5·34-s − 2.13e5·35-s + 3.21e5·37-s − 6.39e5·38-s − 1.07e5·40-s − 1.08e4·41-s + 6.30e5·43-s − 5.49e5·46-s − 4.72e5·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.751·5-s − 1.11·7-s − 0.353·8-s + 0.531·10-s − 0.247·11-s − 0.174·13-s − 0.791·14-s − 1/4·16-s + 1.45·17-s − 2.67·19-s − 0.174·22-s − 1.17·23-s + 25-s − 0.123·26-s + 0.780·29-s − 1.37·31-s + 1.02·34-s − 0.841·35-s + 1.04·37-s − 1.88·38-s − 0.265·40-s − 0.0245·41-s + 1.20·43-s − 0.832·46-s − 0.664·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5151511088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5151511088\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 42 p T - 1361 p^{2} T^{2} - 42 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 1016 T + 208713 T^{2} + 1016 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 1092 T - 18294707 T^{2} + 1092 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 1382 T - 60838593 T^{2} + 1382 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14706 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 39940 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 68712 T + 1316513497 T^{2} + 68712 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 102570 T - 6729271409 T^{2} - 102570 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 227552 T + 24267298593 T^{2} + 227552 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 160526 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10842 T - 194636724917 T^{2} + 10842 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 630748 T + 126024428397 T^{2} - 630748 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 472656 T - 283219426127 T^{2} + 472656 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1494018 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 2640660 T + 4484433750781 T^{2} + 2640660 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 827702 T - 2457652235217 T^{2} + 827702 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 126004 T - 6044834597307 T^{2} - 126004 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1414728 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 980282 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3566800 T - 6481846746159 T^{2} - 3566800 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 5672892 T + 5045652654037 T^{2} + 5672892 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11951190 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8682146 T - 5418625312797 T^{2} + 8682146 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38590282405699067310956022452, −11.06787318355986964097171174062, −10.92875447280971535719004613922, −10.18899675184351352853608829798, −9.861474803056141706691768079576, −9.349678879799478911701259624787, −8.856416775850293193499677398173, −8.118665624761470450068917685764, −7.71774133683985805629707949672, −6.68285425174787471801328566547, −6.48516845401651517589623430707, −5.78086472440448823956642656773, −5.59776990575650498259692524053, −4.44166331907479062758973240146, −4.30458204941907521524887790694, −3.28627334425985638000464321709, −2.86361606282305645118690798344, −2.10434678322277747658240576645, −1.34819969205301651089558808919, −0.16362937463392281351662987648,
0.16362937463392281351662987648, 1.34819969205301651089558808919, 2.10434678322277747658240576645, 2.86361606282305645118690798344, 3.28627334425985638000464321709, 4.30458204941907521524887790694, 4.44166331907479062758973240146, 5.59776990575650498259692524053, 5.78086472440448823956642656773, 6.48516845401651517589623430707, 6.68285425174787471801328566547, 7.71774133683985805629707949672, 8.118665624761470450068917685764, 8.856416775850293193499677398173, 9.349678879799478911701259624787, 9.861474803056141706691768079576, 10.18899675184351352853608829798, 10.92875447280971535719004613922, 11.06787318355986964097171174062, 12.38590282405699067310956022452
