| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 3·11-s − 12-s + 14-s + 16-s + 7·17-s − 18-s + 3·19-s + 21-s + 3·22-s + 23-s + 24-s − 27-s − 28-s − 29-s − 8·31-s − 32-s + 3·33-s − 7·34-s + 36-s − 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 0.688·19-s + 0.218·21-s + 0.639·22-s + 0.208·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.185·29-s − 1.43·31-s − 0.176·32-s + 0.522·33-s − 1.20·34-s + 1/6·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4277643625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4277643625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.91901344041331, −12.63964102085575, −12.26524301705760, −11.63570751770358, −11.28499775807491, −10.72644774321248, −10.23409712566491, −10.01070700503600, −9.376781294615073, −9.054011943740766, −8.305547923946329, −7.825058559889669, −7.373896183010786, −7.119340733548663, −6.336904463854897, −5.795373110583188, −5.434028585053309, −5.035487790049208, −4.184709858708194, −3.524442056837271, −3.031030346433959, −2.502204484890244, −1.534658008783610, −1.197895206076980, −0.2336366706680343,
0.2336366706680343, 1.197895206076980, 1.534658008783610, 2.502204484890244, 3.031030346433959, 3.524442056837271, 4.184709858708194, 5.035487790049208, 5.434028585053309, 5.795373110583188, 6.336904463854897, 7.119340733548663, 7.373896183010786, 7.825058559889669, 8.305547923946329, 9.054011943740766, 9.376781294615073, 10.01070700503600, 10.23409712566491, 10.72644774321248, 11.28499775807491, 11.63570751770358, 12.26524301705760, 12.63964102085575, 12.91901344041331
