| L(s) = 1 | + 4·7-s + 13-s + 6·17-s + 4·19-s − 6·23-s + 6·29-s − 10·31-s − 10·37-s − 6·41-s − 4·43-s − 12·47-s + 9·49-s + 12·53-s + 12·59-s + 10·61-s + 14·67-s + 16·73-s + 8·79-s + 12·83-s + 6·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 1.11·29-s − 1.79·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s + 1.64·53-s + 1.56·59-s + 1.28·61-s + 1.71·67-s + 1.87·73-s + 0.900·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.945921950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.945921950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.16772311742607, −12.51872909218974, −12.07366026308231, −11.72248593444492, −11.37617434039626, −10.78693967965582, −10.24941488913105, −9.930967344682637, −9.404735342046266, −8.635557998287078, −8.223430714464486, −8.061327315976999, −7.413933201028215, −6.900588051880571, −6.421164525201329, −5.449404027271624, −5.286276593008153, −5.069680816002037, −4.109239418147030, −3.588772731863164, −3.311528119801491, −2.166201791400224, −1.920119506457424, −1.206840698578163, −0.6041191570013176,
0.6041191570013176, 1.206840698578163, 1.920119506457424, 2.166201791400224, 3.311528119801491, 3.588772731863164, 4.109239418147030, 5.069680816002037, 5.286276593008153, 5.449404027271624, 6.421164525201329, 6.900588051880571, 7.413933201028215, 8.061327315976999, 8.223430714464486, 8.635557998287078, 9.404735342046266, 9.930967344682637, 10.24941488913105, 10.78693967965582, 11.37617434039626, 11.72248593444492, 12.07366026308231, 12.51872909218974, 13.16772311742607
