| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 5·11-s − 12-s + 15-s + 16-s − 7·17-s − 18-s − 20-s − 5·22-s − 7·23-s + 24-s + 25-s − 27-s − 30-s − 10·31-s − 32-s − 5·33-s + 7·34-s + 36-s + 11·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.223·20-s − 1.06·22-s − 1.45·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.182·30-s − 1.79·31-s − 0.176·32-s − 0.870·33-s + 1.20·34-s + 1/6·36-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.90219818671975, −12.63774661375394, −11.79437926935408, −11.61774989701414, −11.42494668342478, −10.81545837121905, −10.33804099464452, −9.768745952068317, −9.412964225563298, −8.872566692817558, −8.478818215132792, −8.054220399245219, −7.235170883096217, −7.019447763182775, −6.575581624990818, −6.001853738683656, −5.658962530254727, −4.840263170561743, −4.246882573738393, −3.909563661392490, −3.437687418956980, −2.386604011366970, −2.036389173020471, −1.369183481767808, −0.6408677687136538, 0,
0.6408677687136538, 1.369183481767808, 2.036389173020471, 2.386604011366970, 3.437687418956980, 3.909563661392490, 4.246882573738393, 4.840263170561743, 5.658962530254727, 6.001853738683656, 6.575581624990818, 7.019447763182775, 7.235170883096217, 8.054220399245219, 8.478818215132792, 8.872566692817558, 9.412964225563298, 9.768745952068317, 10.33804099464452, 10.81545837121905, 11.42494668342478, 11.61774989701414, 11.79437926935408, 12.63774661375394, 12.90219818671975
