| L(s) = 1 | + 3-s + 2·5-s + 2·7-s − 2·9-s + 4·11-s + 4·13-s + 2·15-s + 7·17-s + 3·19-s + 2·21-s − 25-s − 5·27-s − 4·29-s − 6·31-s + 4·33-s + 4·35-s − 2·37-s + 4·39-s + 6·41-s + 5·43-s − 4·45-s + 10·47-s − 3·49-s + 7·51-s + 8·55-s + 3·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s − 2/3·9-s + 1.20·11-s + 1.10·13-s + 0.516·15-s + 1.69·17-s + 0.688·19-s + 0.436·21-s − 1/5·25-s − 0.962·27-s − 0.742·29-s − 1.07·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s + 0.762·43-s − 0.596·45-s + 1.45·47-s − 3/7·49-s + 0.980·51-s + 1.07·55-s + 0.397·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.128951326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.128951326\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.76710392519079116459069235260, −7.30589302467615292251115621349, −6.19346159312306008386633752274, −5.71178360568493368652544630211, −5.28197907244016654009647093797, −3.95475060756719055793880525680, −3.59116233268286206100484024077, −2.60919885992619439279885370424, −1.67543057973761111632475993474, −1.08933587701227700418322939454,
1.08933587701227700418322939454, 1.67543057973761111632475993474, 2.60919885992619439279885370424, 3.59116233268286206100484024077, 3.95475060756719055793880525680, 5.28197907244016654009647093797, 5.71178360568493368652544630211, 6.19346159312306008386633752274, 7.30589302467615292251115621349, 7.76710392519079116459069235260
