| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s − 2·9-s − 10-s + 2·11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 6·17-s − 2·18-s + 7·19-s − 20-s − 21-s + 2·22-s + 24-s − 4·25-s − 26-s − 5·27-s − 28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 1.60·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.821066387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821066387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.84763908186790, −14.09752899799261, −13.87463555126288, −13.38647180390993, −12.83976272156679, −12.12596084616055, −11.73272137963286, −11.30651001487626, −10.83591708386407, −9.917784778266958, −9.513136653890579, −8.906640719105949, −8.445460338275212, −7.713374506061321, −7.199996352247637, −6.767710655159641, −5.925975087717740, −5.539099228253963, −4.803384916913231, −4.031740234646666, −3.644663124680672, −3.023366918496800, −2.362963351974723, −1.694615864066831, −0.5013486288437098,
0.5013486288437098, 1.694615864066831, 2.362963351974723, 3.023366918496800, 3.644663124680672, 4.031740234646666, 4.803384916913231, 5.539099228253963, 5.925975087717740, 6.767710655159641, 7.199996352247637, 7.713374506061321, 8.445460338275212, 8.906640719105949, 9.513136653890579, 9.917784778266958, 10.83591708386407, 11.30651001487626, 11.73272137963286, 12.12596084616055, 12.83976272156679, 13.38647180390993, 13.87463555126288, 14.09752899799261, 14.84763908186790
