| L(s) = 1 | + 3-s + 2·5-s + 9-s + 5·11-s + 4·13-s + 2·15-s − 5·17-s + 19-s − 23-s − 25-s + 27-s − 4·31-s + 5·33-s − 2·37-s + 4·39-s − 2·41-s + 8·43-s + 2·45-s + 5·47-s − 5·51-s − 2·53-s + 10·55-s + 57-s − 8·59-s − 7·61-s + 8·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.516·15-s − 1.21·17-s + 0.229·19-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.718·31-s + 0.870·33-s − 0.328·37-s + 0.640·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s + 0.729·47-s − 0.700·51-s − 0.274·53-s + 1.34·55-s + 0.132·57-s − 1.04·59-s − 0.896·61-s + 0.992·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.142389046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.142389046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.17188432081836, −12.96025319630979, −12.12538634338215, −11.89983128023114, −11.13089781798096, −10.86192650433083, −10.34257019856364, −9.645190997165940, −9.266738448799898, −8.979329906915941, −8.610989401676465, −7.925066927161920, −7.348830303505397, −6.812184700989549, −6.235915039395018, −6.078955587414958, −5.401641192256439, −4.627483351123564, −4.143658367874718, −3.672528562338196, −3.150012501103681, −2.323635539336403, −1.835646558777868, −1.402716924922295, −0.6212981147312981,
0.6212981147312981, 1.402716924922295, 1.835646558777868, 2.323635539336403, 3.150012501103681, 3.672528562338196, 4.143658367874718, 4.627483351123564, 5.401641192256439, 6.078955587414958, 6.235915039395018, 6.812184700989549, 7.348830303505397, 7.925066927161920, 8.610989401676465, 8.979329906915941, 9.266738448799898, 9.645190997165940, 10.34257019856364, 10.86192650433083, 11.13089781798096, 11.89983128023114, 12.12538634338215, 12.96025319630979, 13.17188432081836
