| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 14-s − 15-s + 16-s + 18-s − 5·19-s − 20-s + 21-s + 22-s + 6·23-s + 24-s + 25-s + 27-s + 28-s + 6·29-s − 30-s + 7·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.14·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.244901874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.244901874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.40536429322352, −13.98962094595147, −13.41557075376643, −12.85628418004389, −12.48266012850328, −11.98105598805238, −11.24283313201936, −11.07104479178422, −10.32136710184857, −9.853453855511433, −9.110802699790899, −8.570360501885955, −8.119264751422032, −7.677573898008368, −6.777239328926633, −6.642561681305827, −5.944528678243926, −4.973509490307400, −4.680995106995617, −4.198897031170433, −3.327268630138377, −3.029828104121119, −2.211941727309778, −1.533779871719101, −0.6835201300615938,
0.6835201300615938, 1.533779871719101, 2.211941727309778, 3.029828104121119, 3.327268630138377, 4.198897031170433, 4.680995106995617, 4.973509490307400, 5.944528678243926, 6.642561681305827, 6.777239328926633, 7.677573898008368, 8.119264751422032, 8.570360501885955, 9.110802699790899, 9.853453855511433, 10.32136710184857, 11.07104479178422, 11.24283313201936, 11.98105598805238, 12.48266012850328, 12.85628418004389, 13.41557075376643, 13.98962094595147, 14.40536429322352
