| L(s) = 1 | + 2·5-s − 7-s − 11-s − 13-s − 5·23-s − 25-s − 29-s + 5·31-s − 2·35-s − 3·37-s + 6·41-s − 43-s − 4·47-s − 6·49-s − 3·53-s − 2·55-s − 11·59-s + 4·61-s − 2·65-s + 8·67-s + 8·71-s − 16·73-s + 77-s − 8·79-s − 9·83-s − 9·89-s + 91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s − 1.04·23-s − 1/5·25-s − 0.185·29-s + 0.898·31-s − 0.338·35-s − 0.493·37-s + 0.937·41-s − 0.152·43-s − 0.583·47-s − 6/7·49-s − 0.412·53-s − 0.269·55-s − 1.43·59-s + 0.512·61-s − 0.248·65-s + 0.977·67-s + 0.949·71-s − 1.87·73-s + 0.113·77-s − 0.900·79-s − 0.987·83-s − 0.953·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−7.82643685460337885644742123807, −7.00551256097532808911161225030, −6.22279506492125066035772059512, −5.77607679767723599901734053614, −4.93549222490940743365500402770, −4.12429085797523547129050798491, −3.12618417368165431724088742044, −2.33841958501325439087561580905, −1.47399144819671628240354483852, 0,
1.47399144819671628240354483852, 2.33841958501325439087561580905, 3.12618417368165431724088742044, 4.12429085797523547129050798491, 4.93549222490940743365500402770, 5.77607679767723599901734053614, 6.22279506492125066035772059512, 7.00551256097532808911161225030, 7.82643685460337885644742123807
