| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s − 2·9-s + 3·10-s + 3·11-s + 12-s + 5·13-s + 14-s − 3·15-s + 16-s + 3·17-s + 2·18-s − 7·19-s − 3·20-s − 21-s − 3·22-s − 24-s + 4·25-s − 5·26-s − 5·27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.60·19-s − 0.670·20-s − 0.218·21-s − 0.639·22-s − 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.557225914344671485105254243669, −8.362881589337464825163849726116, −7.48437907454016562660471261041, −6.54889284187627928271651697581, −5.94034768324310863443756512373, −4.37249102076982832012781008279, −3.61617617486703506111053553219, −2.94450443234815510810975798012, −1.45342223182555401966691413635, 0,
1.45342223182555401966691413635, 2.94450443234815510810975798012, 3.61617617486703506111053553219, 4.37249102076982832012781008279, 5.94034768324310863443756512373, 6.54889284187627928271651697581, 7.48437907454016562660471261041, 8.362881589337464825163849726116, 8.557225914344671485105254243669
