L(s) = 1 | − 2-s + 4-s − 8-s + 5·11-s − 13-s + 16-s − 7·17-s + 7·19-s − 5·22-s − 2·23-s − 5·25-s + 26-s + 9·29-s − 32-s + 7·34-s + 4·37-s − 7·38-s − 4·41-s + 2·43-s + 5·44-s + 2·46-s + 3·47-s + 5·50-s − 52-s − 53-s − 9·58-s − 7·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.50·11-s − 0.277·13-s + 1/4·16-s − 1.69·17-s + 1.60·19-s − 1.06·22-s − 0.417·23-s − 25-s + 0.196·26-s + 1.67·29-s − 0.176·32-s + 1.20·34-s + 0.657·37-s − 1.13·38-s − 0.624·41-s + 0.304·43-s + 0.753·44-s + 0.294·46-s + 0.437·47-s + 0.707·50-s − 0.138·52-s − 0.137·53-s − 1.18·58-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542776944\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542776944\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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
−16.38712123942376, −15.94671091021222, −15.49436484190468, −14.80815550145505, −14.09566354272495, −13.77374792992629, −13.02231274060969, −12.13285807746577, −11.72536207464342, −11.40036561133666, −10.56669366176041, −9.881276492133895, −9.429051639694688, −8.871083905111057, −8.303305564115244, −7.521630797588817, −6.908660085688210, −6.391167288140155, −5.747977535693094, −4.725309934880064, −4.121586643050782, −3.269991836481698, −2.387438951432516, −1.556880874123110, −0.6626860901384381,
0.6626860901384381, 1.556880874123110, 2.387438951432516, 3.269991836481698, 4.121586643050782, 4.725309934880064, 5.747977535693094, 6.391167288140155, 6.908660085688210, 7.521630797588817, 8.303305564115244, 8.871083905111057, 9.429051639694688, 9.881276492133895, 10.56669366176041, 11.40036561133666, 11.72536207464342, 12.13285807746577, 13.02231274060969, 13.77374792992629, 14.09566354272495, 14.80815550145505, 15.49436484190468, 15.94671091021222, 16.38712123942376