| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 2·13-s + 2·15-s − 4·17-s + 19-s + 21-s − 6·23-s − 25-s + 27-s − 6·29-s − 5·31-s + 2·35-s + 7·37-s − 2·39-s − 4·43-s + 2·45-s + 4·47-s − 6·49-s − 4·51-s + 6·53-s + 57-s − 6·59-s − 61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.898·31-s + 0.338·35-s + 1.15·37-s − 0.320·39-s − 0.609·43-s + 0.298·45-s + 0.583·47-s − 6/7·49-s − 0.560·51-s + 0.824·53-s + 0.132·57-s − 0.781·59-s − 0.128·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−16.70210438420941, −16.14657570811327, −15.49780902107896, −14.83870108420009, −14.47128106295111, −13.87514159562936, −13.30411755014796, −12.96024031248126, −12.16148168876361, −11.49245973411346, −10.94077368416524, −10.11129241787270, −9.747186085340173, −9.127093376195889, −8.598897716456005, −7.754481877682224, −7.376806362375180, −6.484539505211486, −5.867649794043310, −5.230127029950648, −4.390855270164591, −3.797469982359466, −2.749903962336979, −2.094162148023534, −1.538707466749101, 0,
1.538707466749101, 2.094162148023534, 2.749903962336979, 3.797469982359466, 4.390855270164591, 5.230127029950648, 5.867649794043310, 6.484539505211486, 7.376806362375180, 7.754481877682224, 8.598897716456005, 9.127093376195889, 9.747186085340173, 10.11129241787270, 10.94077368416524, 11.49245973411346, 12.16148168876361, 12.96024031248126, 13.30411755014796, 13.87514159562936, 14.47128106295111, 14.83870108420009, 15.49780902107896, 16.14657570811327, 16.70210438420941
