| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s − 6·11-s − 12-s + 14-s + 16-s + 3·17-s + 2·18-s − 2·19-s + 21-s + 6·22-s + 24-s + 5·27-s − 28-s + 6·29-s + 4·31-s − 32-s + 6·33-s − 3·34-s − 2·36-s − 7·37-s + 2·38-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 − 1.80·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.458·19-s + 0.218·21-s + 1.27·22-s + 0.204·24-s + 0.962·27-s − 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s − 0.514·34-s − 1/3·36-s − 1.15·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.58543076531539186507707996737, −6.68519917257609817275888561025, −6.22050342677947846490119014336, −5.22470196642237218714598599078, −5.11503905300625867515341835277, −3.71632285273985803925942045989, −2.83043023758736407632723876116, −2.30738287414640841822352268120, −0.885388786186633410385805055803, 0,
0.885388786186633410385805055803, 2.30738287414640841822352268120, 2.83043023758736407632723876116, 3.71632285273985803925942045989, 5.11503905300625867515341835277, 5.22470196642237218714598599078, 6.22050342677947846490119014336, 6.68519917257609817275888561025, 7.58543076531539186507707996737
