| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s − 2·11-s + 2·13-s + 2·15-s − 6·19-s + 2·21-s + 23-s − 25-s + 27-s + 6·29-s − 2·33-s + 4·35-s − 2·37-s + 2·39-s + 41-s + 2·45-s + 8·47-s − 3·49-s + 12·53-s − 4·55-s − 6·57-s + 4·59-s + 6·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.516·15-s − 1.37·19-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.348·33-s + 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.156·41-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 1.64·53-s − 0.539·55-s − 0.794·57-s + 0.520·59-s + 0.768·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.842269070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.842269070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.23249971225192, −12.76642476813073, −12.39343014158413, −11.67023330504633, −11.26157683916702, −10.62955604873682, −10.28468664529693, −10.01208299001298, −9.228752039009097, −8.781981022015794, −8.486750428569285, −7.971053968677484, −7.466090227866043, −6.843694566300227, −6.317209719267997, −5.881555024812241, −5.232170117765451, −4.845539350116527, −4.149819558553665, −3.745106389437348, −2.916866884160940, −2.292240297126774, −2.065555139630674, −1.303209469691248, −0.6058773470076675,
0.6058773470076675, 1.303209469691248, 2.065555139630674, 2.292240297126774, 2.916866884160940, 3.745106389437348, 4.149819558553665, 4.845539350116527, 5.232170117765451, 5.881555024812241, 6.317209719267997, 6.843694566300227, 7.466090227866043, 7.971053968677484, 8.486750428569285, 8.781981022015794, 9.228752039009097, 10.01208299001298, 10.28468664529693, 10.62955604873682, 11.26157683916702, 11.67023330504633, 12.39343014158413, 12.76642476813073, 13.23249971225192
