| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 7-s + 3·8-s + 9-s + 10-s − 12-s − 13-s + 14-s − 15-s − 16-s − 6·17-s − 18-s + 20-s − 21-s + 4·23-s + 3·24-s + 25-s + 26-s + 27-s + 28-s − 6·29-s + 30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7481865194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7481865194\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.19664393735172, −12.84729545527093, −12.49054171343965, −11.77371849644963, −11.13246273858134, −10.80848522751064, −10.38506916491218, −9.719213413277030, −9.289376971675433, −8.884228630656032, −8.683570983462859, −8.026998507575828, −7.407497061916628, −7.145036339200098, −6.747567789454139, −5.727513190261115, −5.436553412895853, −4.548403189964306, −4.241500752408261, −3.779718101360600, −3.077290848377846, −2.405480544346234, −1.860044014105818, −1.047301303897825, −0.3116942963846596,
0.3116942963846596, 1.047301303897825, 1.860044014105818, 2.405480544346234, 3.077290848377846, 3.779718101360600, 4.241500752408261, 4.548403189964306, 5.436553412895853, 5.727513190261115, 6.747567789454139, 7.145036339200098, 7.407497061916628, 8.026998507575828, 8.683570983462859, 8.884228630656032, 9.289376971675433, 9.719213413277030, 10.38506916491218, 10.80848522751064, 11.13246273858134, 11.77371849644963, 12.49054171343965, 12.84729545527093, 13.19664393735172
