| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s + 4·11-s − 13-s − 4·14-s − 16-s + 17-s + 4·22-s − 26-s + 4·28-s − 6·29-s − 4·31-s + 5·32-s + 34-s + 10·37-s + 2·41-s − 4·43-s − 4·44-s + 9·49-s + 52-s − 10·53-s + 12·56-s − 6·58-s + 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 1.20·11-s − 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.242·17-s + 0.852·22-s − 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 0.171·34-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.603·44-s + 9/7·49-s + 0.138·52-s − 1.37·53-s + 1.60·56-s − 0.787·58-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.64100422849867, −14.40645563166611, −13.62120871457784, −13.15621977066384, −12.93448834584926, −12.31874074885312, −11.92004263876033, −11.35048321504550, −10.68939779996145, −9.824909574646205, −9.609039010979293, −9.180692724466491, −8.719705004923730, −7.875869010866360, −7.253680431807374, −6.623255512624524, −6.076418800672442, −5.828709824447976, −4.987296949674145, −4.338956432330609, −3.800732514629996, −3.332946509122016, −2.811759613903147, −1.844968004019015, −0.8073281042990971, 0,
0.8073281042990971, 1.844968004019015, 2.811759613903147, 3.332946509122016, 3.800732514629996, 4.338956432330609, 4.987296949674145, 5.828709824447976, 6.076418800672442, 6.623255512624524, 7.253680431807374, 7.875869010866360, 8.719705004923730, 9.180692724466491, 9.609039010979293, 9.824909574646205, 10.68939779996145, 11.35048321504550, 11.92004263876033, 12.31874074885312, 12.93448834584926, 13.15621977066384, 13.62120871457784, 14.40645563166611, 14.64100422849867
