| L(s) = 1 | + 5-s − 2·7-s − 13-s − 3·17-s + 4·19-s + 3·23-s + 25-s − 3·29-s + 2·31-s − 2·35-s − 10·37-s + 43-s − 6·47-s − 3·49-s + 3·53-s − 6·59-s + 13·61-s − 65-s + 14·67-s − 14·73-s − 11·79-s + 12·83-s − 3·85-s − 12·89-s + 2·91-s + 4·95-s − 4·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s + 0.359·31-s − 0.338·35-s − 1.64·37-s + 0.152·43-s − 0.875·47-s − 3/7·49-s + 0.412·53-s − 0.781·59-s + 1.66·61-s − 0.124·65-s + 1.71·67-s − 1.63·73-s − 1.23·79-s + 1.31·83-s − 0.325·85-s − 1.27·89-s + 0.209·91-s + 0.410·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−12.92392398151429, −12.68281859749311, −11.99280276514276, −11.69514121590044, −10.97142829814416, −10.83275876618929, −9.935836182755638, −9.851773769889113, −9.412709659225365, −8.759324807707621, −8.503425014807708, −7.836026983080256, −7.166902031343954, −6.864565351724141, −6.532643121855558, −5.752353222617114, −5.459435148664099, −4.917171536422725, −4.349968637937016, −3.700126928980353, −3.161297982058180, −2.775508451387595, −2.036824131927846, −1.531427017304249, −0.7193653640543006, 0,
0.7193653640543006, 1.531427017304249, 2.036824131927846, 2.775508451387595, 3.161297982058180, 3.700126928980353, 4.349968637937016, 4.917171536422725, 5.459435148664099, 5.752353222617114, 6.532643121855558, 6.864565351724141, 7.166902031343954, 7.836026983080256, 8.503425014807708, 8.759324807707621, 9.412709659225365, 9.851773769889113, 9.935836182755638, 10.83275876618929, 10.97142829814416, 11.69514121590044, 11.99280276514276, 12.68281859749311, 12.92392398151429
