| L(s) = 1 | + 3-s + 4·7-s + 9-s + 13-s − 2·17-s + 8·19-s + 4·21-s + 4·23-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 39-s − 6·41-s + 4·43-s + 4·47-s + 9·49-s − 2·51-s + 6·53-s + 8·57-s + 8·59-s + 10·61-s + 4·63-s − 4·67-s + 4·69-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 1.05·57-s + 1.04·59-s + 1.28·61-s + 0.503·63-s − 0.488·67-s + 0.481·69-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.226061570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.226061570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.29704839500457, −13.71875440862263, −13.48284298378062, −12.88480863857393, −12.14348757329769, −11.62621116794627, −11.30057897891857, −10.86372552238619, −10.07065066550584, −9.692892925422773, −9.022675562090735, −8.546350547609295, −8.138342467385082, −7.482128116016887, −7.229790462262849, −6.495326001269788, −5.574974792027579, −5.293730584191700, −4.598220564066052, −4.107077168121648, −3.396141281486262, −2.689854507216347, −2.114711629786990, −1.288162784951851, −0.8445083024729771,
0.8445083024729771, 1.288162784951851, 2.114711629786990, 2.689854507216347, 3.396141281486262, 4.107077168121648, 4.598220564066052, 5.293730584191700, 5.574974792027579, 6.495326001269788, 7.229790462262849, 7.482128116016887, 8.138342467385082, 8.546350547609295, 9.022675562090735, 9.692892925422773, 10.07065066550584, 10.86372552238619, 11.30057897891857, 11.62621116794627, 12.14348757329769, 12.88480863857393, 13.48284298378062, 13.71875440862263, 14.29704839500457
