| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s − 2·9-s − 11-s − 12-s − 2·13-s − 14-s + 16-s + 3·17-s + 2·18-s − 19-s − 21-s + 22-s − 6·23-s + 24-s + 2·26-s + 5·27-s + 28-s − 9·29-s + 5·31-s − 32-s + 33-s − 3·34-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 − 2/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.218·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 0.392·26-s + 0.962·27-s + 0.188·28-s − 1.67·29-s + 0.898·31-s − 0.176·32-s + 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−10.28756550502077574077550197198, −9.635083365431788819695943976404, −8.433889912409664065106990618200, −7.86951967459801418659648715702, −6.74649729638384385602961829144, −5.77539302433093715120788386982, −4.92393270075562316789294902433, −3.32416571879742266865701460608, −1.88765560992081383778636536806, 0,
1.88765560992081383778636536806, 3.32416571879742266865701460608, 4.92393270075562316789294902433, 5.77539302433093715120788386982, 6.74649729638384385602961829144, 7.86951967459801418659648715702, 8.433889912409664065106990618200, 9.635083365431788819695943976404, 10.28756550502077574077550197198
