| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 11-s + 13-s + 16-s + 4·17-s − 4·19-s − 2·20-s − 22-s + 5·23-s − 25-s + 26-s − 6·29-s − 11·31-s + 32-s + 4·34-s + 7·37-s − 4·38-s − 2·40-s + 3·41-s + 2·43-s − 44-s + 5·46-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.447·20-s − 0.213·22-s + 1.04·23-s − 1/5·25-s + 0.196·26-s − 1.11·29-s − 1.97·31-s + 0.176·32-s + 0.685·34-s + 1.15·37-s − 0.648·38-s − 0.316·40-s + 0.468·41-s + 0.304·43-s − 0.150·44-s + 0.737·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−16.67164811684774, −16.09784167511873, −15.48093875174896, −14.86518602105063, −14.69662509985105, −13.92319978032833, −13.02949476244209, −12.90207636622200, −12.22633722543358, −11.55439772683221, −11.01445166777504, −10.69061724167804, −9.749158121081690, −9.089864289315469, −8.381970792476990, −7.528541755142556, −7.441816557427568, −6.490846451045097, −5.686023264738040, −5.277481384340610, −4.258885983753305, −3.872277580247143, −3.148228541898551, −2.302018815819095, −1.265498616089948, 0,
1.265498616089948, 2.302018815819095, 3.148228541898551, 3.872277580247143, 4.258885983753305, 5.277481384340610, 5.686023264738040, 6.490846451045097, 7.441816557427568, 7.528541755142556, 8.381970792476990, 9.089864289315469, 9.749158121081690, 10.69061724167804, 11.01445166777504, 11.55439772683221, 12.22633722543358, 12.90207636622200, 13.02949476244209, 13.92319978032833, 14.69662509985105, 14.86518602105063, 15.48093875174896, 16.09784167511873, 16.67164811684774
