| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s − 2·9-s + 10-s + 6·11-s − 12-s + 7·13-s + 15-s + 16-s − 17-s + 2·18-s − 5·19-s − 20-s − 6·22-s + 6·23-s + 24-s + 25-s − 7·26-s + 5·27-s + 3·29-s − 30-s − 5·31-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 − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 1.94·13-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 1.14·19-s − 0.223·20-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 1.37·26-s + 0.962·27-s + 0.557·29-s − 0.182·30-s − 0.898·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282688193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282688193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−8.030218370137044929724896659917, −6.87920890994416866552949192982, −6.49121340945934493945649811858, −6.05391921649844972412930385332, −5.13545086664598873141803332436, −4.01993485882416420320295950736, −3.67277250929244509135001349569, −2.55957877217888359169250522155, −1.35590575573606746687173259385, −0.72607315381442123622310730492,
0.72607315381442123622310730492, 1.35590575573606746687173259385, 2.55957877217888359169250522155, 3.67277250929244509135001349569, 4.01993485882416420320295950736, 5.13545086664598873141803332436, 6.05391921649844972412930385332, 6.49121340945934493945649811858, 6.87920890994416866552949192982, 8.030218370137044929724896659917
