| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s − 2·11-s + 2·12-s − 2·13-s − 14-s + 16-s − 17-s + 18-s − 2·21-s − 2·22-s + 4·23-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s − 28-s + 4·29-s + 32-s − 4·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.436·21-s − 0.426·22-s + 0.834·23-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s − 0.188·28-s + 0.742·29-s + 0.176·32-s − 0.696·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317877218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317877218\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.40266974818482445145259685126, −11.31891938439764243607274530051, −10.14752828221237751819967846664, −9.211602639697435722792087458225, −8.118421380469541351780580189944, −7.24873510617453755225057715176, −5.94586048673065741845555135065, −4.61890591955424914588423851483, −3.30389600119444587052090687147, −2.36560540929594190713679838208,
2.36560540929594190713679838208, 3.30389600119444587052090687147, 4.61890591955424914588423851483, 5.94586048673065741845555135065, 7.24873510617453755225057715176, 8.118421380469541351780580189944, 9.211602639697435722792087458225, 10.14752828221237751819967846664, 11.31891938439764243607274530051, 12.40266974818482445145259685126
