| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 6·13-s + 2·15-s + 6·17-s − 4·19-s + 2·21-s − 2·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 4·35-s − 8·37-s + 6·39-s + 6·41-s + 2·45-s − 2·47-s − 3·49-s + 6·51-s − 12·53-s − 4·57-s + 8·59-s + 8·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.436·21-s − 0.417·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.676·35-s − 1.31·37-s + 0.960·39-s + 0.937·41-s + 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.840·51-s − 1.64·53-s − 0.529·57-s + 1.04·59-s + 1.02·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.82353237980443, −12.59780530927908, −11.62696167749135, −11.50192599286448, −10.79903417049814, −10.59260384165929, −9.977380945603189, −9.583414677514413, −9.066612156634193, −8.700393274643172, −8.179822820489052, −7.825102028517722, −7.388508568025130, −6.636960884980212, −6.270451914826463, −5.663340985554293, −5.414656777904574, −4.867295271617175, −3.956255812387772, −3.763019109120481, −3.311166974284998, −2.456553619177948, −1.904705529276042, −1.568440150507119, −1.068456903897590, 0,
1.068456903897590, 1.568440150507119, 1.904705529276042, 2.456553619177948, 3.311166974284998, 3.763019109120481, 3.956255812387772, 4.867295271617175, 5.414656777904574, 5.663340985554293, 6.270451914826463, 6.636960884980212, 7.388508568025130, 7.825102028517722, 8.179822820489052, 8.700393274643172, 9.066612156634193, 9.583414677514413, 9.977380945603189, 10.59260384165929, 10.79903417049814, 11.50192599286448, 11.62696167749135, 12.59780530927908, 12.82353237980443
