| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 4·13-s + 21-s − 4·23-s + 27-s + 2·29-s − 4·31-s − 33-s + 8·37-s + 4·39-s − 2·41-s − 8·43-s − 12·47-s + 49-s − 4·53-s − 12·59-s + 2·61-s + 63-s + 4·67-s − 4·69-s − 8·71-s − 12·73-s − 77-s − 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.218·21-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.549·53-s − 1.56·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s − 0.481·69-s − 0.949·71-s − 1.40·73-s − 0.113·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.73042109897710, −15.11652983684850, −14.68898617756087, −14.12367274183507, −13.59576870386162, −13.11285040415659, −12.67912441244945, −11.82650500157983, −11.43397738933561, −10.79630244564945, −10.25488393484934, −9.640634785115160, −9.087768014358602, −8.350322682794498, −8.101817665451943, −7.487446938570076, −6.671490719047542, −6.156904338269559, −5.470378168404055, −4.692707887069200, −4.122664541960705, −3.379399779812400, −2.812343470519211, −1.831007777627419, −1.317644994040928, 0,
1.317644994040928, 1.831007777627419, 2.812343470519211, 3.379399779812400, 4.122664541960705, 4.692707887069200, 5.470378168404055, 6.156904338269559, 6.671490719047542, 7.487446938570076, 8.101817665451943, 8.350322682794498, 9.087768014358602, 9.640634785115160, 10.25488393484934, 10.79630244564945, 11.43397738933561, 11.82650500157983, 12.67912441244945, 13.11285040415659, 13.59576870386162, 14.12367274183507, 14.68898617756087, 15.11652983684850, 15.73042109897710
