| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 7-s + 8-s + 9-s + 3·10-s − 11-s + 12-s + 4·13-s + 14-s + 3·15-s + 16-s + 18-s + 3·20-s + 21-s − 22-s + 8·23-s + 24-s + 4·25-s + 4·26-s + 27-s + 28-s − 29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.235·18-s + 0.670·20-s + 0.218·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 0.185·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.918302772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.918302772\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−16.39685270281133, −15.60587120274212, −14.99983473025684, −14.68818941848778, −13.92774584144323, −13.54402242262254, −13.12166109394062, −12.74793342116757, −11.77821869717031, −11.17726545607402, −10.58053570125726, −10.09827239169486, −9.259202391639076, −8.878368394114606, −8.149226883635928, −7.393020092630504, −6.674917688448702, −6.137364363503654, −5.379637480025929, −4.975139519872507, −4.033595474168197, −3.265221332261065, −2.593607791999252, −1.802311219555041, −1.152377560514915,
1.152377560514915, 1.802311219555041, 2.593607791999252, 3.265221332261065, 4.033595474168197, 4.975139519872507, 5.379637480025929, 6.137364363503654, 6.674917688448702, 7.393020092630504, 8.149226883635928, 8.878368394114606, 9.259202391639076, 10.09827239169486, 10.58053570125726, 11.17726545607402, 11.77821869717031, 12.74793342116757, 13.12166109394062, 13.54402242262254, 13.92774584144323, 14.68818941848778, 14.99983473025684, 15.60587120274212, 16.39685270281133
