| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 3·11-s + 2·14-s + 16-s − 2·19-s + 3·22-s + 3·23-s − 5·25-s − 2·28-s + 6·29-s + 4·31-s − 32-s − 2·37-s + 2·38-s + 8·43-s − 3·44-s − 3·46-s + 9·47-s − 3·49-s + 5·50-s + 6·53-s + 2·56-s − 6·58-s − 9·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s + 0.534·14-s + 1/4·16-s − 0.458·19-s + 0.639·22-s + 0.625·23-s − 25-s − 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.328·37-s + 0.324·38-s + 1.21·43-s − 0.452·44-s − 0.442·46-s + 1.31·47-s − 3/7·49-s + 0.707·50-s + 0.824·53-s + 0.267·56-s − 0.787·58-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.59229754603172806869961489222, −6.69682386357905054764056692281, −6.20910312443316778751036993919, −5.45892860882435724261370714959, −4.61434286249165636724700559408, −3.70792061530075869618196268496, −2.82188365665973309838641226268, −2.28072213369652358890896789741, −1.02656732134574195310140915123, 0,
1.02656732134574195310140915123, 2.28072213369652358890896789741, 2.82188365665973309838641226268, 3.70792061530075869618196268496, 4.61434286249165636724700559408, 5.45892860882435724261370714959, 6.20910312443316778751036993919, 6.69682386357905054764056692281, 7.59229754603172806869961489222
