| L(s) = 1 | − 2-s − 3·3-s + 4-s + 5-s + 3·6-s − 8-s + 6·9-s − 10-s − 4·11-s − 3·12-s + 3·13-s − 3·15-s + 16-s − 17-s − 6·18-s − 3·19-s + 20-s + 4·22-s − 6·23-s + 3·24-s + 25-s − 3·26-s − 9·27-s + 9·29-s + 3·30-s + 3·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.316·10-s − 1.20·11-s − 0.866·12-s + 0.832·13-s − 0.774·15-s + 1/4·16-s − 0.242·17-s − 1.41·18-s − 0.688·19-s + 0.223·20-s + 0.852·22-s − 1.25·23-s + 0.612·24-s + 1/5·25-s − 0.588·26-s − 1.73·27-s + 1.67·29-s + 0.547·30-s + 0.538·31-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−7.41202181544195242338587099006, −6.54069201317224559079984060982, −6.12364568696473731368238663487, −5.64597237858296012193640577872, −4.80748200066536741636268565663, −4.17047612738883826915177392024, −2.85719514653559564758193860695, −1.90672158329285742203290786482, −0.936828998266211293288446184145, 0,
0.936828998266211293288446184145, 1.90672158329285742203290786482, 2.85719514653559564758193860695, 4.17047612738883826915177392024, 4.80748200066536741636268565663, 5.64597237858296012193640577872, 6.12364568696473731368238663487, 6.54069201317224559079984060982, 7.41202181544195242338587099006
