| L(s) = 1 | − 2-s + 4-s − 2·5-s + 4·7-s − 8-s + 2·10-s + 2·11-s − 13-s − 4·14-s + 16-s − 3·19-s − 2·20-s − 2·22-s + 4·23-s − 25-s + 26-s + 4·28-s + 29-s + 10·31-s − 32-s − 8·35-s − 3·37-s + 3·38-s + 2·40-s + 9·41-s + 10·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 0.353·8-s + 0.632·10-s + 0.603·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.688·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.185·29-s + 1.79·31-s − 0.176·32-s − 1.35·35-s − 0.493·37-s + 0.486·38-s + 0.316·40-s + 1.40·41-s + 1.52·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119660481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119660481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−10.68299358198908874533045184178, −9.426239317853153498818497806462, −8.597337122054794303887071926779, −7.916330845798711819639209835252, −7.29605795256860414714946161697, −6.15656799324274914391120744904, −4.82781001276383869188938424008, −4.03427014532869057908561192700, −2.46543461143783178701717494619, −1.05596748914819227302692975266,
1.05596748914819227302692975266, 2.46543461143783178701717494619, 4.03427014532869057908561192700, 4.82781001276383869188938424008, 6.15656799324274914391120744904, 7.29605795256860414714946161697, 7.916330845798711819639209835252, 8.597337122054794303887071926779, 9.426239317853153498818497806462, 10.68299358198908874533045184178
