| L(s) = 1 | − 3-s − 5-s − 2·9-s − 2·11-s − 4·13-s + 15-s − 17-s − 6·19-s − 23-s + 25-s + 5·27-s − 29-s + 6·31-s + 2·33-s − 2·37-s + 4·39-s − 5·41-s + 9·43-s + 2·45-s − 12·47-s + 51-s + 4·53-s + 2·55-s + 6·57-s − 8·59-s + 15·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s − 0.242·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s + 1.07·31-s + 0.348·33-s − 0.328·37-s + 0.640·39-s − 0.780·41-s + 1.37·43-s + 0.298·45-s − 1.75·47-s + 0.140·51-s + 0.549·53-s + 0.269·55-s + 0.794·57-s − 1.04·59-s + 1.92·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.01975799915119, −12.76014087158918, −12.31671047396729, −11.81573730204921, −11.44938821592912, −11.03890464648763, −10.48195126809944, −10.15241120096530, −9.706563593190647, −8.875686547023970, −8.701491907295702, −7.986793408383220, −7.808759797918937, −7.067283777971914, −6.571954574048608, −6.266561966434071, −5.546784832102919, −5.153791115124813, −4.661322523034308, −4.224260665302853, −3.558632097059873, −2.816377269908763, −2.497957759310870, −1.863292080687845, −0.9352543326301144, 0, 0,
0.9352543326301144, 1.863292080687845, 2.497957759310870, 2.816377269908763, 3.558632097059873, 4.224260665302853, 4.661322523034308, 5.153791115124813, 5.546784832102919, 6.266561966434071, 6.571954574048608, 7.067283777971914, 7.808759797918937, 7.986793408383220, 8.701491907295702, 8.875686547023970, 9.706563593190647, 10.15241120096530, 10.48195126809944, 11.03890464648763, 11.44938821592912, 11.81573730204921, 12.31671047396729, 12.76014087158918, 13.01975799915119
