| L(s) = 1 | − 2-s − 4-s − 4·7-s + 3·8-s − 3·11-s − 13-s + 4·14-s − 16-s − 2·17-s + 3·22-s + 3·23-s + 26-s + 4·28-s + 31-s − 5·32-s + 2·34-s + 10·37-s − 9·41-s − 43-s + 3·44-s − 3·46-s + 5·47-s + 9·49-s + 52-s + 53-s − 12·56-s − 59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.51·7-s + 1.06·8-s − 0.904·11-s − 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.639·22-s + 0.625·23-s + 0.196·26-s + 0.755·28-s + 0.179·31-s − 0.883·32-s + 0.342·34-s + 1.64·37-s − 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.442·46-s + 0.729·47-s + 9/7·49-s + 0.138·52-s + 0.137·53-s − 1.60·56-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 43 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−7.50692021599299708555965947057, −6.72974300617864097529050895282, −6.13052836459108541751253637736, −5.22687049962670872617582221609, −4.62242978189937898508501818568, −3.73209181949160162591589512479, −2.99144331688627586872022512268, −2.17296875465962166006828400425, −0.830539591099400868994010661150, 0,
0.830539591099400868994010661150, 2.17296875465962166006828400425, 2.99144331688627586872022512268, 3.73209181949160162591589512479, 4.62242978189937898508501818568, 5.22687049962670872617582221609, 6.13052836459108541751253637736, 6.72974300617864097529050895282, 7.50692021599299708555965947057
