| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s − 2·11-s + 2·13-s + 2·14-s − 16-s − 2·17-s + 19-s + 2·22-s − 2·26-s + 2·28-s + 6·29-s − 4·31-s − 5·32-s + 2·34-s − 2·37-s − 38-s − 2·41-s + 10·43-s + 2·44-s − 3·49-s − 2·52-s + 10·53-s − 6·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 0.603·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.426·22-s − 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.328·37-s − 0.162·38-s − 0.312·41-s + 1.52·43-s + 0.301·44-s − 3/7·49-s − 0.277·52-s + 1.37·53-s − 0.801·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−8.150551111650656185785582528931, −7.44238016003126383703095707917, −6.70194368061493157602962019022, −5.84978808578604366759073708414, −5.03934857222197027243114791763, −4.20598633363177245161796831395, −3.39079330112971162956218222284, −2.36012356875242254611192023051, −1.10979871829357564694162994016, 0,
1.10979871829357564694162994016, 2.36012356875242254611192023051, 3.39079330112971162956218222284, 4.20598633363177245161796831395, 5.03934857222197027243114791763, 5.84978808578604366759073708414, 6.70194368061493157602962019022, 7.44238016003126383703095707917, 8.150551111650656185785582528931
