| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s + 7-s − 8-s + 9-s + 4·10-s + 12-s − 13-s − 14-s − 4·15-s + 16-s + 2·17-s − 18-s + 7·19-s − 4·20-s + 21-s + 6·23-s − 24-s + 11·25-s + 26-s + 27-s + 28-s + 3·29-s + 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.60·19-s − 0.894·20-s + 0.218·21-s + 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097683255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097683255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−14.41862755433970, −13.91104448185942, −13.01916178284850, −12.57123977397471, −12.11909844172617, −11.55442376184125, −11.24700723131367, −10.74678872860937, −10.09226372898091, −9.523451419373215, −8.923715996243050, −8.561586863757919, −8.007673948004783, −7.491636199478349, −7.208639990296859, −6.866794069283428, −5.680777797666242, −5.217885399843955, −4.479246750592321, −3.899628686333531, −3.285453358055344, −2.897915285777798, −2.045872458598098, −0.9442118144120260, −0.7083696280707199,
0.7083696280707199, 0.9442118144120260, 2.045872458598098, 2.897915285777798, 3.285453358055344, 3.899628686333531, 4.479246750592321, 5.217885399843955, 5.680777797666242, 6.866794069283428, 7.208639990296859, 7.491636199478349, 8.007673948004783, 8.561586863757919, 8.923715996243050, 9.523451419373215, 10.09226372898091, 10.74678872860937, 11.24700723131367, 11.55442376184125, 12.11909844172617, 12.57123977397471, 13.01916178284850, 13.91104448185942, 14.41862755433970
