| L(s) = 1 | − 2·3-s − 5-s + 3·7-s + 9-s + 2·15-s + 6·17-s − 6·21-s − 23-s − 4·25-s + 4·27-s + 3·29-s + 10·31-s − 3·35-s − 2·37-s + 7·41-s − 43-s − 45-s − 4·47-s + 2·49-s − 12·51-s + 6·53-s − 5·59-s + 11·61-s + 3·63-s − 67-s + 2·69-s + 16·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.516·15-s + 1.45·17-s − 1.30·21-s − 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.557·29-s + 1.79·31-s − 0.507·35-s − 0.328·37-s + 1.09·41-s − 0.152·43-s − 0.149·45-s − 0.583·47-s + 2/7·49-s − 1.68·51-s + 0.824·53-s − 0.650·59-s + 1.40·61-s + 0.377·63-s − 0.122·67-s + 0.240·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
| 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
−12.56675615888937, −12.13317759536728, −11.94252214785768, −11.49077177484391, −11.19387761628992, −10.58964540125998, −10.30072960803895, −9.734604234337026, −9.330227329498338, −8.376680414761283, −8.174382694378943, −7.959443196176491, −7.195434852219513, −6.801424621813864, −6.187351026118108, −5.726448674251658, −5.327695419169873, −4.895828860646336, −4.357639704720495, −3.953619328644403, −3.197040529224166, −2.646234923904665, −1.922468407648957, −1.125369086681763, −0.8706000642604717, 0,
0.8706000642604717, 1.125369086681763, 1.922468407648957, 2.646234923904665, 3.197040529224166, 3.953619328644403, 4.357639704720495, 4.895828860646336, 5.327695419169873, 5.726448674251658, 6.187351026118108, 6.801424621813864, 7.195434852219513, 7.959443196176491, 8.174382694378943, 8.376680414761283, 9.330227329498338, 9.734604234337026, 10.30072960803895, 10.58964540125998, 11.19387761628992, 11.49077177484391, 11.94252214785768, 12.13317759536728, 12.56675615888937
