| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 11-s + 2·13-s − 15-s − 2·17-s + 4·19-s − 4·21-s + 25-s − 27-s − 2·29-s − 33-s + 4·35-s − 2·37-s − 2·39-s − 6·41-s + 45-s + 8·47-s + 9·49-s + 2·51-s − 2·53-s + 55-s − 4·57-s + 4·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.174·33-s + 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.883823192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883823192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.632399140620866568052624915702, −8.821871286475972723014693180634, −8.009566062492333632833163401416, −7.17665204953591422349485387256, −6.24069064176806043982657250764, −5.34851746170090479318781098943, −4.75162937320343842221229075126, −3.67014648079352463457080576580, −2.11052861860361450952195837922, −1.14739005074193094296989473725,
1.14739005074193094296989473725, 2.11052861860361450952195837922, 3.67014648079352463457080576580, 4.75162937320343842221229075126, 5.34851746170090479318781098943, 6.24069064176806043982657250764, 7.17665204953591422349485387256, 8.009566062492333632833163401416, 8.821871286475972723014693180634, 9.632399140620866568052624915702
