| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 12-s + 13-s − 2·14-s + 15-s + 16-s − 18-s + 2·19-s + 20-s + 2·21-s − 6·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s − 30-s + 8·31-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 + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 0.182·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.405857374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.405857374\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−11.20291413168661963945398212907, −10.16327219341387316664345665196, −9.534301462999349112263247556576, −8.437246301408132287185658098919, −7.923933777636678991733597967480, −6.77145317316863409022172537659, −5.65253747259908769204779494812, −4.28133784539552676878054432798, −2.75082196840239005573414536478, −1.49299392430960775577654396970,
1.49299392430960775577654396970, 2.75082196840239005573414536478, 4.28133784539552676878054432798, 5.65253747259908769204779494812, 6.77145317316863409022172537659, 7.923933777636678991733597967480, 8.437246301408132287185658098919, 9.534301462999349112263247556576, 10.16327219341387316664345665196, 11.20291413168661963945398212907
