| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 4·11-s − 4·13-s + 2·15-s + 2·17-s + 8·19-s + 2·21-s − 4·23-s − 25-s + 27-s − 8·29-s − 4·31-s + 4·33-s + 4·35-s − 2·37-s − 4·39-s + 4·43-s + 2·45-s − 2·47-s − 3·49-s + 2·51-s + 4·53-s + 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 0.640·39-s + 0.609·43-s + 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.280·51-s + 0.549·53-s + 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.07641101718160, −12.29695176282082, −11.85277634053374, −11.63876134593990, −11.13775386216721, −10.38543432793186, −9.893230895483046, −9.729152409626407, −9.170643643701822, −8.945417870842590, −8.199105733567918, −7.717842072797465, −7.248730154261638, −7.096794555939884, −6.156209885985271, −5.837360856637851, −5.189481651648915, −5.024050322411060, −4.112322981037920, −3.777758297825445, −3.229280010149720, −2.471303035655914, −2.042564351919384, −1.488413178865771, −1.089059708239147, 0,
1.089059708239147, 1.488413178865771, 2.042564351919384, 2.471303035655914, 3.229280010149720, 3.777758297825445, 4.112322981037920, 5.024050322411060, 5.189481651648915, 5.837360856637851, 6.156209885985271, 7.096794555939884, 7.248730154261638, 7.717842072797465, 8.199105733567918, 8.945417870842590, 9.170643643701822, 9.729152409626407, 9.893230895483046, 10.38543432793186, 11.13775386216721, 11.63876134593990, 11.85277634053374, 12.29695176282082, 13.07641101718160
