| L(s) = 1 | − 2·5-s + 5·11-s − 6·13-s − 4·17-s − 4·19-s − 4·23-s − 25-s − 7·29-s + 3·31-s + 8·37-s − 6·41-s − 8·43-s − 6·47-s + 6·53-s − 10·55-s − 7·59-s + 12·65-s − 10·67-s − 4·71-s − 13·73-s + 3·79-s + 7·83-s + 8·85-s + 6·89-s + 8·95-s + 5·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.50·11-s − 1.66·13-s − 0.970·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.29·29-s + 0.538·31-s + 1.31·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s + 0.824·53-s − 1.34·55-s − 0.911·59-s + 1.48·65-s − 1.22·67-s − 0.474·71-s − 1.52·73-s + 0.337·79-s + 0.768·83-s + 0.867·85-s + 0.635·89-s + 0.820·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5829806509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5829806509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.28790628716713, −15.10819683854333, −14.65947010945951, −14.10964030468793, −13.29867295413833, −12.90405094190902, −12.02884902666850, −11.81692376884816, −11.45872030406007, −10.65577855870987, −9.963052241835496, −9.503922161686731, −8.848878041368904, −8.336430770090567, −7.552872730830053, −7.229611547708099, −6.421248659872812, −6.061025088670834, −4.925372259742631, −4.443338633177203, −3.960541082127978, −3.235849289681603, −2.252766118960333, −1.660522171701295, −0.2999870183126519,
0.2999870183126519, 1.660522171701295, 2.252766118960333, 3.235849289681603, 3.960541082127978, 4.443338633177203, 4.925372259742631, 6.061025088670834, 6.421248659872812, 7.229611547708099, 7.552872730830053, 8.336430770090567, 8.848878041368904, 9.503922161686731, 9.963052241835496, 10.65577855870987, 11.45872030406007, 11.81692376884816, 12.02884902666850, 12.90405094190902, 13.29867295413833, 14.10964030468793, 14.65947010945951, 15.10819683854333, 15.28790628716713
