| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 6·11-s − 7·13-s + 16-s + 17-s − 2·20-s − 6·22-s + 23-s − 25-s − 7·26-s − 2·29-s − 4·31-s + 32-s + 34-s + 10·37-s − 2·40-s − 5·41-s − 43-s − 6·44-s + 46-s + 4·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 1.80·11-s − 1.94·13-s + 1/4·16-s + 0.242·17-s − 0.447·20-s − 1.27·22-s + 0.208·23-s − 1/5·25-s − 1.37·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 1.64·37-s − 0.316·40-s − 0.780·41-s − 0.152·43-s − 0.904·44-s + 0.147·46-s + 0.583·47-s − 49-s − 0.141·50-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.29699999825143, −13.20772777814032, −12.54839245349145, −12.31088016952759, −11.75449489027353, −11.25970081789080, −10.89601540310881, −10.18973700773768, −9.923332850027845, −9.403490714303672, −8.564751292835030, −8.030543541757155, −7.613712181901468, −7.348462768184921, −6.887316376385973, −6.001706316931711, −5.511568296869710, −5.065345054977171, −4.603139486030015, −4.151240362723449, −3.349798980124847, −2.904330448028133, −2.377095079183023, −1.845542247204088, −0.6139339446003913, 0,
0.6139339446003913, 1.845542247204088, 2.377095079183023, 2.904330448028133, 3.349798980124847, 4.151240362723449, 4.603139486030015, 5.065345054977171, 5.511568296869710, 6.001706316931711, 6.887316376385973, 7.348462768184921, 7.613712181901468, 8.030543541757155, 8.564751292835030, 9.403490714303672, 9.923332850027845, 10.18973700773768, 10.89601540310881, 11.25970081789080, 11.75449489027353, 12.31088016952759, 12.54839245349145, 13.20772777814032, 13.29699999825143
