| 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
| \(L(4)\) |
\(\approx\) |
\(1.046643207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046643207\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 8682146 T - 5418625312797 T^{2} + 8682146 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.65341764430539733528704565279, −11.10355986587216814180902079944, −10.71218291818596138449705124557, −10.47569540525007078525289469244, −9.638795621049765180799968260241, −9.066311200632443772938439629593, −8.790405544267505999981052644115, −8.531816145197165378502381217038, −7.48658467146709452281922416170, −7.25703396239742252937867622676, −6.52952453752052166835363067049, −6.27389069334316184509517122646, −5.32016729676995239998335383465, −4.51159413349610679811054862942, −4.00002256728181171364686269065, −3.55194764576111035388758874877, −2.27859775762293972197089846409, −2.26055892583048259342003969450, −0.62624570706144068709864117372, −0.53688727646775178878984435345,
0.53688727646775178878984435345, 0.62624570706144068709864117372, 2.26055892583048259342003969450, 2.27859775762293972197089846409, 3.55194764576111035388758874877, 4.00002256728181171364686269065, 4.51159413349610679811054862942, 5.32016729676995239998335383465, 6.27389069334316184509517122646, 6.52952453752052166835363067049, 7.25703396239742252937867622676, 7.48658467146709452281922416170, 8.531816145197165378502381217038, 8.790405544267505999981052644115, 9.066311200632443772938439629593, 9.638795621049765180799968260241, 10.47569540525007078525289469244, 10.71218291818596138449705124557, 11.10355986587216814180902079944, 11.65341764430539733528704565279
