| L(s) = 1 | + 2-s + 3·3-s − 4-s + 3·6-s − 4·7-s − 3·8-s + 6·9-s − 3·12-s + 3·13-s − 4·14-s − 16-s − 3·17-s + 6·18-s − 12·21-s + 7·23-s − 9·24-s − 5·25-s + 3·26-s + 9·27-s + 4·28-s + 7·29-s − 4·31-s + 5·32-s − 3·34-s − 6·36-s − 5·37-s + 9·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s − 1.51·7-s − 1.06·8-s + 2·9-s − 0.866·12-s + 0.832·13-s − 1.06·14-s − 1/4·16-s − 0.727·17-s + 1.41·18-s − 2.61·21-s + 1.45·23-s − 1.83·24-s − 25-s + 0.588·26-s + 1.73·27-s + 0.755·28-s + 1.29·29-s − 0.718·31-s + 0.883·32-s − 0.514·34-s − 36-s − 0.821·37-s + 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 \) | |
| 53 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.63072633506469, −15.47410897657005, −14.77222411947639, −14.28870415883488, −13.61812788987323, −13.30102079290486, −13.16777618774714, −12.44584186122014, −11.91142356316452, −10.90793669020852, −10.19437403924409, −9.653501285432257, −9.160972736992145, −8.765865469740393, −8.319833892641558, −7.497791602963804, −6.626804129892306, −6.473767706969329, −5.428844507075824, −4.711477256315458, −3.810488035302603, −3.618436953905972, −2.976496890534693, −2.436267950661955, −1.299911459431236, 0,
1.299911459431236, 2.436267950661955, 2.976496890534693, 3.618436953905972, 3.810488035302603, 4.711477256315458, 5.428844507075824, 6.473767706969329, 6.626804129892306, 7.497791602963804, 8.319833892641558, 8.765865469740393, 9.160972736992145, 9.653501285432257, 10.19437403924409, 10.90793669020852, 11.91142356316452, 12.44584186122014, 13.16777618774714, 13.30102079290486, 13.61812788987323, 14.28870415883488, 14.77222411947639, 15.47410897657005, 15.63072633506469
