| L(s) = 1 | − 3·7-s − 3·11-s − 4·13-s + 4·17-s + 2·29-s − 3·31-s − 8·41-s − 12·43-s + 12·47-s + 2·49-s − 5·53-s + 12·67-s − 12·71-s − 9·73-s + 9·77-s − 12·79-s − 9·83-s − 4·89-s + 12·91-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 12·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.904·11-s − 1.10·13-s + 0.970·17-s + 0.371·29-s − 0.538·31-s − 1.24·41-s − 1.82·43-s + 1.75·47-s + 2/7·49-s − 0.686·53-s + 1.46·67-s − 1.42·71-s − 1.05·73-s + 1.02·77-s − 1.35·79-s − 0.987·83-s − 0.423·89-s + 1.25·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.10·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.52404224739125, −13.16335420886845, −12.78858349620944, −12.16975833990300, −12.02883233394839, −11.37279029321139, −10.67736917063903, −10.17134668796551, −10.02284456412229, −9.496573177816829, −8.992822043303273, −8.302978792384821, −7.956344304648643, −7.295338247037117, −6.939909269105565, −6.495702996750870, −5.640745779642531, −5.483424663015773, −4.861728117624728, −4.221786652166760, −3.568913028088538, −2.935064281323570, −2.752805654570145, −1.893961149708801, −1.169259814980723, 0, 0,
1.169259814980723, 1.893961149708801, 2.752805654570145, 2.935064281323570, 3.568913028088538, 4.221786652166760, 4.861728117624728, 5.483424663015773, 5.640745779642531, 6.495702996750870, 6.939909269105565, 7.295338247037117, 7.956344304648643, 8.302978792384821, 8.992822043303273, 9.496573177816829, 10.02284456412229, 10.17134668796551, 10.67736917063903, 11.37279029321139, 12.02883233394839, 12.16975833990300, 12.78858349620944, 13.16335420886845, 13.52404224739125
