| L(s) = 1 | + 2·7-s − 3·11-s + 2·13-s + 3·17-s − 19-s − 6·23-s − 6·29-s + 4·31-s − 4·37-s + 9·41-s + 43-s − 6·47-s − 3·49-s + 12·53-s + 3·59-s − 8·61-s − 5·67-s + 12·71-s − 11·73-s − 6·77-s + 4·79-s − 12·83-s + 6·89-s + 4·91-s − 5·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.904·11-s + 0.554·13-s + 0.727·17-s − 0.229·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.657·37-s + 1.40·41-s + 0.152·43-s − 0.875·47-s − 3/7·49-s + 1.64·53-s + 0.390·59-s − 1.02·61-s − 0.610·67-s + 1.42·71-s − 1.28·73-s − 0.683·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045325834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045325834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.56830013285616, −12.92835269724918, −12.64912152950746, −11.90153884401594, −11.66035396293894, −11.01166513582182, −10.66385482057384, −10.08753299307848, −9.758243928592953, −9.016698919319610, −8.513610316281102, −8.079935930036910, −7.610756868689446, −7.282385097936016, −6.403639809806397, −5.938225207877947, −5.480476788748889, −4.962220034774766, −4.320761502462819, −3.831446458823491, −3.188449705524099, −2.486251790312180, −1.912320951406751, −1.295069530384788, −0.4355864740550435,
0.4355864740550435, 1.295069530384788, 1.912320951406751, 2.486251790312180, 3.188449705524099, 3.831446458823491, 4.320761502462819, 4.962220034774766, 5.480476788748889, 5.938225207877947, 6.403639809806397, 7.282385097936016, 7.610756868689446, 8.079935930036910, 8.513610316281102, 9.016698919319610, 9.758243928592953, 10.08753299307848, 10.66385482057384, 11.01166513582182, 11.66035396293894, 11.90153884401594, 12.64912152950746, 12.92835269724918, 13.56830013285616
