| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 4·13-s − 2·15-s + 2·17-s − 8·19-s − 4·23-s − 25-s − 27-s − 8·29-s + 4·31-s − 4·33-s + 2·37-s − 4·39-s + 41-s − 4·43-s + 2·45-s − 2·47-s − 2·51-s + 4·53-s + 8·55-s + 8·57-s + 12·59-s + 6·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s + 0.156·41-s − 0.609·43-s + 0.298·45-s − 0.291·47-s − 0.280·51-s + 0.549·53-s + 1.07·55-s + 1.05·57-s + 1.56·59-s + 0.768·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401217202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401217202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.62600987445954, −13.26704655654112, −13.00909606398855, −12.17077465151099, −11.89174930949125, −11.33598572654897, −10.84779914590048, −10.28894794304446, −9.990869979308482, −9.164052071172855, −9.090409079886616, −8.234677825093875, −7.921848084786535, −6.937983051570834, −6.608081774394145, −6.016542859598399, −5.870636774069610, −5.185326512693236, −4.329420341094184, −3.968996897003155, −3.473693239302634, −2.416908115187489, −1.837896990851689, −1.385501984823422, −0.5121006908931679,
0.5121006908931679, 1.385501984823422, 1.837896990851689, 2.416908115187489, 3.473693239302634, 3.968996897003155, 4.329420341094184, 5.185326512693236, 5.870636774069610, 6.016542859598399, 6.608081774394145, 6.937983051570834, 7.921848084786535, 8.234677825093875, 9.090409079886616, 9.164052071172855, 9.990869979308482, 10.28894794304446, 10.84779914590048, 11.33598572654897, 11.89174930949125, 12.17077465151099, 13.00909606398855, 13.26704655654112, 13.62600987445954
