| L(s) = 1 | − 3-s + 3·5-s − 4·7-s + 9-s + 4·11-s − 3·15-s + 3·17-s − 4·19-s + 4·21-s − 8·23-s + 4·25-s − 27-s − 5·29-s − 8·31-s − 4·33-s − 12·35-s + 7·37-s − 9·41-s + 8·43-s + 3·45-s − 4·47-s + 9·49-s − 3·51-s − 5·53-s + 12·55-s + 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.774·15-s + 0.727·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s + 4/5·25-s − 0.192·27-s − 0.928·29-s − 1.43·31-s − 0.696·33-s − 2.02·35-s + 1.15·37-s − 1.40·41-s + 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.420·51-s − 0.686·53-s + 1.61·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.068841354782045634951774537598, −7.00325665069443133718092842768, −6.39755459516855968192164438757, −5.98286646106290729095599009411, −5.42773694312854151272894191913, −4.12481282660569680151485111057, −3.52879366434702914029135828412, −2.31591186510545533778177435596, −1.46978367895275451110919687433, 0,
1.46978367895275451110919687433, 2.31591186510545533778177435596, 3.52879366434702914029135828412, 4.12481282660569680151485111057, 5.42773694312854151272894191913, 5.98286646106290729095599009411, 6.39755459516855968192164438757, 7.00325665069443133718092842768, 8.068841354782045634951774537598
