| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 11-s − 2·13-s − 15-s − 17-s − 4·19-s + 4·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 33-s − 4·35-s + 10·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 12·47-s + 9·49-s − 51-s − 6·53-s − 55-s − 4·57-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 − 0.242·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.405158415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.405158415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.99702551563404, −12.78393326400453, −12.29099533697039, −11.59748958765635, −11.24060120589456, −11.02462776284268, −10.37203754514530, −9.652320295152282, −9.495285211726257, −8.581910556605443, −8.366860803533097, −8.070549798403722, −7.432000139025129, −7.026540987459051, −6.462091823794340, −5.808165905724799, −5.098800851239294, −4.701608110690507, −4.251679907055313, −3.791031959594476, −3.018052709314163, −2.398901991015293, −1.901221244965494, −1.309439081753693, −0.5121118283465890,
0.5121118283465890, 1.309439081753693, 1.901221244965494, 2.398901991015293, 3.018052709314163, 3.791031959594476, 4.251679907055313, 4.701608110690507, 5.098800851239294, 5.808165905724799, 6.462091823794340, 7.026540987459051, 7.432000139025129, 8.070549798403722, 8.366860803533097, 8.581910556605443, 9.495285211726257, 9.652320295152282, 10.37203754514530, 11.02462776284268, 11.24060120589456, 11.59748958765635, 12.29099533697039, 12.78393326400453, 12.99702551563404
