| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 12-s + 2·13-s − 14-s − 16-s − 2·17-s − 18-s − 6·19-s + 21-s − 8·23-s + 3·24-s − 5·25-s − 2·26-s + 27-s − 28-s + 6·29-s − 2·31-s − 5·32-s + 2·34-s − 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.218·21-s − 1.66·23-s + 0.612·24-s − 25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 127 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.443906215241180126857704990058, −8.117248107156138680135515694087, −7.18478634495540751678064887408, −6.30167790101972141651105237708, −5.30496434478245662145579871450, −4.22163802006655008928110832197, −3.87576444481729557861693659494, −2.35115524823199716633752036871, −1.52276924969993744607094913618, 0,
1.52276924969993744607094913618, 2.35115524823199716633752036871, 3.87576444481729557861693659494, 4.22163802006655008928110832197, 5.30496434478245662145579871450, 6.30167790101972141651105237708, 7.18478634495540751678064887408, 8.117248107156138680135515694087, 8.443906215241180126857704990058
