| L(s) = 1 | + 5-s + 4·7-s − 2·11-s − 6·13-s + 2·17-s − 5·19-s + 4·23-s − 4·25-s + 10·29-s + 5·31-s + 4·35-s − 10·37-s − 2·41-s + 4·43-s + 12·47-s + 9·49-s + 11·53-s − 2·55-s − 5·59-s − 6·65-s − 5·67-s + 12·71-s − 3·73-s − 8·77-s + 17·79-s − 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.603·11-s − 1.66·13-s + 0.485·17-s − 1.14·19-s + 0.834·23-s − 4/5·25-s + 1.85·29-s + 0.898·31-s + 0.676·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s + 1.51·53-s − 0.269·55-s − 0.650·59-s − 0.744·65-s − 0.610·67-s + 1.42·71-s − 0.351·73-s − 0.911·77-s + 1.91·79-s − 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.161057497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.161057497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.85689789934292, −12.26872807433846, −12.10206575988382, −11.70684065490136, −10.88820897096713, −10.64947125865739, −10.10055248860932, −9.942468671992941, −8.962461495730092, −8.804279862376006, −8.141285469364424, −7.797311378427010, −7.289537785436236, −6.846896398798517, −6.199132111161527, −5.560894003554000, −5.116172175061922, −4.755151216248659, −4.374522131379937, −3.621514116392164, −2.735184971904456, −2.368664733494270, −1.967661722696504, −1.162061554713608, −0.5094289984169784,
0.5094289984169784, 1.162061554713608, 1.967661722696504, 2.368664733494270, 2.735184971904456, 3.621514116392164, 4.374522131379937, 4.755151216248659, 5.116172175061922, 5.560894003554000, 6.199132111161527, 6.846896398798517, 7.289537785436236, 7.797311378427010, 8.141285469364424, 8.804279862376006, 8.962461495730092, 9.942468671992941, 10.10055248860932, 10.64947125865739, 10.88820897096713, 11.70684065490136, 12.10206575988382, 12.26872807433846, 12.85689789934292
