| L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s − 7-s + 8-s + 6·9-s + 10-s + 6·11-s + 3·12-s + 13-s − 14-s + 3·15-s + 16-s + 6·17-s + 6·18-s − 3·19-s + 20-s − 3·21-s + 6·22-s + 3·24-s − 4·25-s + 26-s + 9·27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 1.80·11-s + 0.866·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.45·17-s + 1.41·18-s − 0.688·19-s + 0.223·20-s − 0.654·21-s + 1.27·22-s + 0.612·24-s − 4/5·25-s + 0.196·26-s + 1.73·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.11959910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.11959910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.69961621221689, −14.18360573701223, −13.91206754839152, −13.50260337149677, −12.85661236496827, −12.32832063132430, −11.91762941531823, −11.28732463405588, −10.24899623456144, −10.05742984698397, −9.452005382528115, −8.977851472860247, −8.342961203724718, −8.001399510547420, −7.157757075482496, −6.659584903309365, −6.247474298400224, −5.465273867541661, −4.611685295571103, −4.001081190949605, −3.446687219007421, −3.195646669367295, −2.252722707416592, −1.693509036872152, −1.063924209561131,
1.063924209561131, 1.693509036872152, 2.252722707416592, 3.195646669367295, 3.446687219007421, 4.001081190949605, 4.611685295571103, 5.465273867541661, 6.247474298400224, 6.659584903309365, 7.157757075482496, 8.001399510547420, 8.342961203724718, 8.977851472860247, 9.452005382528115, 10.05742984698397, 10.24899623456144, 11.28732463405588, 11.91762941531823, 12.32832063132430, 12.85661236496827, 13.50260337149677, 13.91206754839152, 14.18360573701223, 14.69961621221689
