| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s + 2·5-s − 6·6-s + 7-s + 6·9-s + 4·10-s + 5·11-s − 6·12-s + 2·14-s − 6·15-s − 4·16-s + 12·18-s + 4·20-s − 3·21-s + 10·22-s + 2·23-s − 25-s − 9·27-s + 2·28-s + 6·29-s − 12·30-s + 4·31-s − 8·32-s − 15·33-s + 2·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 4-s + 0.894·5-s − 2.44·6-s + 0.377·7-s + 2·9-s + 1.26·10-s + 1.50·11-s − 1.73·12-s + 0.534·14-s − 1.54·15-s − 16-s + 2.82·18-s + 0.894·20-s − 0.654·21-s + 2.13·22-s + 0.417·23-s − 1/5·25-s − 1.73·27-s + 0.377·28-s + 1.11·29-s − 2.19·30-s + 0.718·31-s − 1.41·32-s − 2.61·33-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6253 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.320944196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.320944196\) |
| \(L(\frac{3}{2})\) |
|
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 | 13 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−7.63656655642374522700520176619, −6.73307583483590975174974584066, −6.26400357550309171652320691735, −5.93044736055536875701719168406, −5.22356134270092187334177881398, −4.49940339852377717970941068220, −4.15997403211195474184227824915, −2.95391348418649775215944249318, −1.78793685065005709454099470196, −0.887657337582636960036233148138,
0.887657337582636960036233148138, 1.78793685065005709454099470196, 2.95391348418649775215944249318, 4.15997403211195474184227824915, 4.49940339852377717970941068220, 5.22356134270092187334177881398, 5.93044736055536875701719168406, 6.26400357550309171652320691735, 6.73307583483590975174974584066, 7.63656655642374522700520176619
