| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 12-s + 2·14-s − 15-s + 16-s − 5·17-s − 18-s + 5·19-s + 20-s + 2·21-s − 8·23-s + 24-s − 4·25-s − 27-s − 2·28-s + 2·29-s + 30-s − 3·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.436·21-s − 1.66·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.182·30-s − 0.538·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3559250574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3559250574\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.70016612020170, −12.85762143508111, −12.62596315680709, −11.94556430567553, −11.56083604422711, −11.19107776560173, −10.36633312945364, −10.19716008567212, −9.721119858884954, −9.209292484440180, −8.804608505286481, −8.088397521519080, −7.640250145361535, −7.013756827233911, −6.526335129684826, −6.212833481286941, −5.540991312434047, −5.172905320013973, −4.342615411494422, −3.691721994461198, −3.230098966138076, −2.223661867576686, −1.996265879252411, −1.122475325886453, −0.2235206016851431,
0.2235206016851431, 1.122475325886453, 1.996265879252411, 2.223661867576686, 3.230098966138076, 3.691721994461198, 4.342615411494422, 5.172905320013973, 5.540991312434047, 6.212833481286941, 6.526335129684826, 7.013756827233911, 7.640250145361535, 8.088397521519080, 8.804608505286481, 9.209292484440180, 9.721119858884954, 10.19716008567212, 10.36633312945364, 11.19107776560173, 11.56083604422711, 11.94556430567553, 12.62596315680709, 12.85762143508111, 13.70016612020170
