| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 2·13-s + 15-s + 2·17-s + 4·21-s + 25-s − 27-s − 6·29-s + 8·31-s + 4·35-s + 2·37-s − 2·39-s − 6·41-s − 8·43-s − 45-s + 8·47-s + 9·49-s − 2·51-s − 10·53-s − 12·59-s + 2·61-s − 4·63-s − 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s − 1.56·59-s + 0.256·61-s − 0.503·63-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 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
−12.79977950401346, −12.30186483168343, −11.79227076956841, −11.66095970413526, −10.80596174365167, −10.64732403449409, −10.04957923114132, −9.622071699854688, −9.300678700895418, −8.693216897304856, −8.170254885398333, −7.672160401321169, −7.191147751486330, −6.569826963802577, −6.393015274987322, −5.851691722415574, −5.397805682143019, −4.707621368745895, −4.290099268520717, −3.547524869692453, −3.338780240934005, −2.801646291683011, −1.982092798655302, −1.279908563039541, −0.5935948892369118, 0,
0.5935948892369118, 1.279908563039541, 1.982092798655302, 2.801646291683011, 3.338780240934005, 3.547524869692453, 4.290099268520717, 4.707621368745895, 5.397805682143019, 5.851691722415574, 6.393015274987322, 6.569826963802577, 7.191147751486330, 7.672160401321169, 8.170254885398333, 8.693216897304856, 9.300678700895418, 9.622071699854688, 10.04957923114132, 10.64732403449409, 10.80596174365167, 11.66095970413526, 11.79227076956841, 12.30186483168343, 12.79977950401346
