| L(s) = 1 | + 3·7-s + 4·11-s + 3·17-s + 8·19-s − 2·29-s − 4·31-s − 8·37-s + 9·41-s + 6·43-s + 7·47-s + 2·49-s + 6·53-s + 12·59-s + 10·61-s − 10·67-s − 4·71-s + 9·73-s + 12·77-s + 5·79-s − 6·83-s − 13·89-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 1.20·11-s + 0.727·17-s + 1.83·19-s − 0.371·29-s − 0.718·31-s − 1.31·37-s + 1.40·41-s + 0.914·43-s + 1.02·47-s + 2/7·49-s + 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.22·67-s − 0.474·71-s + 1.05·73-s + 1.36·77-s + 0.562·79-s − 0.658·83-s − 1.37·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.614642525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.614642525\) |
| \(L(\frac{3}{2})\) |
|
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.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.69589701935747, −12.97922570063808, −12.38876768530852, −11.92899935309253, −11.66874426893690, −11.12053079515944, −10.74307822947850, −10.00129117646284, −9.625079582250503, −9.002004288539844, −8.759572511918150, −7.980862889795984, −7.582829559340829, −7.144152966534435, −6.677182761736233, −5.702273682426719, −5.540897121713841, −5.048265234963642, −4.161926438125951, −3.922618141922834, −3.245779425526653, −2.521912332660114, −1.777392410538877, −1.205853948167361, −0.7350079125051868,
0.7350079125051868, 1.205853948167361, 1.777392410538877, 2.521912332660114, 3.245779425526653, 3.922618141922834, 4.161926438125951, 5.048265234963642, 5.540897121713841, 5.702273682426719, 6.677182761736233, 7.144152966534435, 7.582829559340829, 7.980862889795984, 8.759572511918150, 9.002004288539844, 9.625079582250503, 10.00129117646284, 10.74307822947850, 11.12053079515944, 11.66874426893690, 11.92899935309253, 12.38876768530852, 12.97922570063808, 13.69589701935747
