| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s − 2·9-s + 2·10-s + 5·11-s − 12-s − 14-s + 2·15-s + 16-s + 2·17-s + 2·18-s − 4·19-s − 2·20-s − 21-s − 5·22-s − 9·23-s + 24-s − 25-s + 5·27-s + 28-s − 2·30-s + 5·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s + 1.50·11-s − 0.288·12-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s − 1.06·22-s − 1.87·23-s + 0.204·24-s − 1/5·25-s + 0.962·27-s + 0.188·28-s − 0.365·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−8.523235559506182904045176798221, −7.967237699788896987138434023119, −7.16118816607441415874990006022, −6.21461681414132879150838049679, −5.78859923248525809812017303512, −4.37187378077028315346850081311, −3.85951608502984751539598244017, −2.53264614515264394499990069674, −1.24612752292527170450284146924, 0,
1.24612752292527170450284146924, 2.53264614515264394499990069674, 3.85951608502984751539598244017, 4.37187378077028315346850081311, 5.78859923248525809812017303512, 6.21461681414132879150838049679, 7.16118816607441415874990006022, 7.967237699788896987138434023119, 8.523235559506182904045176798221
