| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 12-s − 2·13-s − 2·14-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·21-s − 6·23-s + 24-s − 2·26-s + 27-s − 2·28-s − 2·29-s + 4·31-s + 32-s − 6·34-s + 36-s − 4·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.66891106935231975432778817820, −6.73158882555610470481961011753, −6.44243257144511312198352421748, −5.45037301781696306661697712981, −4.68876018134468850150298084088, −3.96683303978095929746346330965, −3.21382796641085520523967299454, −2.51044006152578818866073835576, −1.65520847522090394002717268097, 0,
1.65520847522090394002717268097, 2.51044006152578818866073835576, 3.21382796641085520523967299454, 3.96683303978095929746346330965, 4.68876018134468850150298084088, 5.45037301781696306661697712981, 6.44243257144511312198352421748, 6.73158882555610470481961011753, 7.66891106935231975432778817820
