| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 4·11-s + 4·13-s + 2·15-s + 2·17-s − 8·19-s − 2·21-s − 4·23-s − 25-s − 27-s + 8·29-s − 4·31-s + 4·33-s − 4·35-s + 2·37-s − 4·39-s − 4·43-s − 2·45-s − 2·47-s − 3·49-s − 2·51-s − 4·53-s + 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.609·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s − 0.280·51-s − 0.549·53-s + 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5223589519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5223589519\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.20126917003086, −13.34280890730310, −12.86420099256068, −12.56999980409280, −11.86566432970409, −11.51237080600924, −10.99675693590734, −10.57209974937291, −10.27311433176968, −9.537234094955681, −8.670688225794315, −8.316912278066533, −7.917788059152851, −7.562306854668692, −6.684353900051832, −6.227003757735676, −5.766524688728691, −4.931861227103234, −4.672046671102563, −3.980262531230053, −3.511268995847408, −2.659868107395893, −1.923806270594474, −1.256506091734745, −0.2532899113479254,
0.2532899113479254, 1.256506091734745, 1.923806270594474, 2.659868107395893, 3.511268995847408, 3.980262531230053, 4.672046671102563, 4.931861227103234, 5.766524688728691, 6.227003757735676, 6.684353900051832, 7.562306854668692, 7.917788059152851, 8.316912278066533, 8.670688225794315, 9.537234094955681, 10.27311433176968, 10.57209974937291, 10.99675693590734, 11.51237080600924, 11.86566432970409, 12.56999980409280, 12.86420099256068, 13.34280890730310, 14.20126917003086
