| L(s) = 1 | + 7-s + 5·11-s − 4·13-s + 4·17-s + 8·19-s − 8·23-s + 2·29-s − 7·31-s − 8·37-s − 4·43-s + 4·47-s − 6·49-s + 7·53-s − 12·67-s + 4·71-s + 15·73-s + 5·77-s − 4·79-s − 83-s − 12·89-s − 4·91-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.50·11-s − 1.10·13-s + 0.970·17-s + 1.83·19-s − 1.66·23-s + 0.371·29-s − 1.25·31-s − 1.31·37-s − 0.609·43-s + 0.583·47-s − 6/7·49-s + 0.961·53-s − 1.46·67-s + 0.474·71-s + 1.75·73-s + 0.569·77-s − 0.450·79-s − 0.109·83-s − 1.27·89-s − 0.419·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 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.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.79492258600273, −12.82418011396084, −12.22747085804628, −12.07513002307512, −11.71126391870373, −11.23220693912732, −10.50675870165772, −10.02421674842914, −9.572811729285984, −9.361681712339879, −8.621988257925583, −8.150242744133137, −7.562707519705388, −7.209819522844322, −6.755573874594996, −6.060660084037291, −5.458302756916915, −5.227591405028132, −4.473072095903445, −3.892758510293891, −3.457645554752851, −2.891343025646744, −1.990875482359053, −1.570594837519900, −0.9236876438054423, 0,
0.9236876438054423, 1.570594837519900, 1.990875482359053, 2.891343025646744, 3.457645554752851, 3.892758510293891, 4.473072095903445, 5.227591405028132, 5.458302756916915, 6.060660084037291, 6.755573874594996, 7.209819522844322, 7.562707519705388, 8.150242744133137, 8.621988257925583, 9.361681712339879, 9.572811729285984, 10.02421674842914, 10.50675870165772, 11.23220693912732, 11.71126391870373, 12.07513002307512, 12.22747085804628, 12.82418011396084, 13.79492258600273
