| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 4·7-s − 4·10-s − 3·11-s + 7·13-s − 8·14-s − 4·16-s − 17-s − 4·19-s + 4·20-s + 6·22-s − 23-s − 25-s − 14·26-s + 8·28-s + 9·29-s − 2·31-s + 8·32-s + 2·34-s + 8·35-s − 8·37-s + 8·38-s + 9·41-s + 7·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 1.51·7-s − 1.26·10-s − 0.904·11-s + 1.94·13-s − 2.13·14-s − 16-s − 0.242·17-s − 0.917·19-s + 0.894·20-s + 1.27·22-s − 0.208·23-s − 1/5·25-s − 2.74·26-s + 1.51·28-s + 1.67·29-s − 0.359·31-s + 1.41·32-s + 0.342·34-s + 1.35·35-s − 1.31·37-s + 1.29·38-s + 1.40·41-s + 1.06·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9552159907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9552159907\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−10.80010067496362414705646758430, −10.26491053566870849867075963176, −9.074783363565654434917333220654, −8.400667580128532101109124055567, −7.897941236349574718941751767418, −6.58720795425678370738564517992, −5.54281389578195108069602057654, −4.33407792608305306521972331424, −2.25417992666271586330169990313, −1.28016298028817331562348462813,
1.28016298028817331562348462813, 2.25417992666271586330169990313, 4.33407792608305306521972331424, 5.54281389578195108069602057654, 6.58720795425678370738564517992, 7.897941236349574718941751767418, 8.400667580128532101109124055567, 9.074783363565654434917333220654, 10.26491053566870849867075963176, 10.80010067496362414705646758430
