| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s + 12-s + 13-s − 4·14-s + 16-s + 6·17-s − 18-s − 4·19-s + 4·21-s + 4·23-s − 24-s − 5·25-s − 26-s + 27-s + 4·28-s − 6·29-s + 10·31-s − 32-s − 6·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + 1.79·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 2939 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.29466196825493, −12.56569916417000, −12.19162151081339, −11.63666676564268, −11.25558373580923, −10.85830775678631, −10.28868713137851, −9.924662471507946, −9.309905906256375, −8.971710438866300, −8.279235184450776, −8.064427785548524, −7.677038971054286, −7.330124718004915, −6.443682855706268, −6.149354618910621, −5.432609462740520, −4.906793146348390, −4.418645654444236, −3.816155351392576, −3.151040448208309, −2.661623928841560, −1.895281309291410, −1.490701675917008, −1.028889501608966, 0,
1.028889501608966, 1.490701675917008, 1.895281309291410, 2.661623928841560, 3.151040448208309, 3.816155351392576, 4.418645654444236, 4.906793146348390, 5.432609462740520, 6.149354618910621, 6.443682855706268, 7.330124718004915, 7.677038971054286, 8.064427785548524, 8.279235184450776, 8.971710438866300, 9.309905906256375, 9.924662471507946, 10.28868713137851, 10.85830775678631, 11.25558373580923, 11.63666676564268, 12.19162151081339, 12.56569916417000, 13.29466196825493
