| L(s) = 1 | − 3·3-s + 4·5-s + 3·7-s + 6·9-s + 3·11-s + 6·13-s − 12·15-s − 4·17-s + 6·19-s − 9·21-s − 6·23-s + 11·25-s − 9·27-s − 4·29-s − 9·33-s + 12·35-s − 37-s − 18·39-s − 5·41-s − 6·43-s + 24·45-s − 9·47-s + 2·49-s + 12·51-s + 5·53-s + 12·55-s − 18·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.13·7-s + 2·9-s + 0.904·11-s + 1.66·13-s − 3.09·15-s − 0.970·17-s + 1.37·19-s − 1.96·21-s − 1.25·23-s + 11/5·25-s − 1.73·27-s − 0.742·29-s − 1.56·33-s + 2.02·35-s − 0.164·37-s − 2.88·39-s − 0.780·41-s − 0.914·43-s + 3.57·45-s − 1.31·47-s + 2/7·49-s + 1.68·51-s + 0.686·53-s + 1.61·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678346953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678346953\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.912789757820576242145457882489, −9.196930623939619111723766168561, −8.171116173413720852515064409422, −6.70215534378769498619674725507, −6.37130663081964170851026344460, −5.50386762915442186383450566417, −5.07368852733938728543400465493, −3.89119998683865980446546080420, −1.80714683000640168781195911291, −1.25838069591403157091036114514,
1.25838069591403157091036114514, 1.80714683000640168781195911291, 3.89119998683865980446546080420, 5.07368852733938728543400465493, 5.50386762915442186383450566417, 6.37130663081964170851026344460, 6.70215534378769498619674725507, 8.171116173413720852515064409422, 9.196930623939619111723766168561, 9.912789757820576242145457882489
