| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 4·11-s + 12-s − 13-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 4·22-s − 8·23-s + 24-s + 25-s − 26-s + 27-s − 10·29-s − 30-s + 32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.85·29-s − 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606032422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606032422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−15.66075382693484, −15.14284518432649, −14.51900778352019, −14.12445126692240, −13.66039533275035, −12.92485772610442, −12.50799686424913, −12.07246509428157, −11.35535292058912, −10.80632537028046, −10.12421305843508, −9.656810329346160, −9.036915061513299, −7.926118273243647, −7.743603137171598, −7.506807951828579, −6.435272216091755, −5.684335548851780, −5.282412623656170, −4.535550377995135, −3.641742851397186, −3.403014495492179, −2.477772322445215, −1.875409607697586, −0.6641227802026693,
0.6641227802026693, 1.875409607697586, 2.477772322445215, 3.403014495492179, 3.641742851397186, 4.535550377995135, 5.282412623656170, 5.684335548851780, 6.435272216091755, 7.506807951828579, 7.743603137171598, 7.926118273243647, 9.036915061513299, 9.656810329346160, 10.12421305843508, 10.80632537028046, 11.35535292058912, 12.07246509428157, 12.50799686424913, 12.92485772610442, 13.66039533275035, 14.12445126692240, 14.51900778352019, 15.14284518432649, 15.66075382693484
