| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 6·11-s − 2·14-s + 16-s + 5·17-s + 2·20-s + 6·22-s − 23-s − 25-s − 2·28-s − 29-s + 7·31-s + 32-s + 5·34-s − 4·35-s − 37-s + 2·40-s − 2·41-s − 43-s + 6·44-s − 46-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 1.80·11-s − 0.534·14-s + 1/4·16-s + 1.21·17-s + 0.447·20-s + 1.27·22-s − 0.208·23-s − 1/5·25-s − 0.377·28-s − 0.185·29-s + 1.25·31-s + 0.176·32-s + 0.857·34-s − 0.676·35-s − 0.164·37-s + 0.316·40-s − 0.312·41-s − 0.152·43-s + 0.904·44-s − 0.147·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.472997335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.472997335\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.45103641158083, −12.97509903145767, −12.29301144878499, −12.03555503092199, −11.66244212253849, −11.06385334593183, −10.35762284383807, −9.936321281588727, −9.583431053145184, −9.229369838380187, −8.471517463344177, −8.033853309115750, −7.239182295706729, −6.784486403966677, −6.322097766757821, −6.015056557057651, −5.490887347038417, −4.864958251707949, −4.240808233852248, −3.624378498015489, −3.320166097479205, −2.608457211037503, −1.874849530917670, −1.372384405528714, −0.6766521531326978,
0.6766521531326978, 1.372384405528714, 1.874849530917670, 2.608457211037503, 3.320166097479205, 3.624378498015489, 4.240808233852248, 4.864958251707949, 5.490887347038417, 6.015056557057651, 6.322097766757821, 6.784486403966677, 7.239182295706729, 8.033853309115750, 8.471517463344177, 9.229369838380187, 9.583431053145184, 9.936321281588727, 10.35762284383807, 11.06385334593183, 11.66244212253849, 12.03555503092199, 12.29301144878499, 12.97509903145767, 13.45103641158083
