| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 3·7-s − 8-s + 9-s + 3·11-s − 2·12-s − 3·13-s − 3·14-s + 16-s − 18-s − 4·19-s − 6·21-s − 3·22-s − 5·23-s + 2·24-s + 3·26-s + 4·27-s + 3·28-s − 8·29-s + 8·31-s − 32-s − 6·33-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 1.30·21-s − 0.639·22-s − 1.04·23-s + 0.408·24-s + 0.588·26-s + 0.769·27-s + 0.566·28-s − 1.48·29-s + 1.43·31-s − 0.176·32-s − 1.04·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−16.43189941766397, −16.27062991567703, −15.18722400709714, −14.78776926374448, −14.43046366384558, −13.58319486314297, −12.84282609760330, −11.95194132801382, −11.87245972463387, −11.37822462681473, −10.71541246090020, −10.26209708224561, −9.589243954018644, −8.919451657335170, −8.211359859176399, −7.804198639255558, −6.926657679725888, −6.471269960368070, −5.778790928945308, −5.181620924860497, −4.467947069371919, −3.842076911262664, −2.530483199151533, −1.846545042940031, −0.9750166843295595, 0,
0.9750166843295595, 1.846545042940031, 2.530483199151533, 3.842076911262664, 4.467947069371919, 5.181620924860497, 5.778790928945308, 6.471269960368070, 6.926657679725888, 7.804198639255558, 8.211359859176399, 8.919451657335170, 9.589243954018644, 10.26209708224561, 10.71541246090020, 11.37822462681473, 11.87245972463387, 11.95194132801382, 12.84282609760330, 13.58319486314297, 14.43046366384558, 14.78776926374448, 15.18722400709714, 16.27062991567703, 16.43189941766397
