| L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 3·9-s − 2·10-s − 4·11-s − 2·13-s + 14-s + 16-s + 2·17-s − 3·18-s − 4·19-s − 2·20-s − 4·22-s − 25-s − 2·26-s + 28-s − 6·29-s + 32-s + 2·34-s − 2·35-s − 3·36-s − 6·37-s − 4·38-s − 2·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s − 0.648·38-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.443511495069253015977599687403, −8.243652201107820206424774957135, −7.88361489297860101098679345997, −6.96896340283507601308979022106, −5.76268875084749510894360042644, −5.14917934793066143835530572952, −4.15785893596527821113679099960, −3.17432866378691893362933092869, −2.18911649830885718412641508923, 0,
2.18911649830885718412641508923, 3.17432866378691893362933092869, 4.15785893596527821113679099960, 5.14917934793066143835530572952, 5.76268875084749510894360042644, 6.96896340283507601308979022106, 7.88361489297860101098679345997, 8.243652201107820206424774957135, 9.443511495069253015977599687403
