| L(s) = 1 | − 3·5-s − 4·7-s − 2·13-s + 6·17-s + 2·19-s − 3·23-s + 4·25-s + 6·29-s − 8·31-s + 12·35-s + 2·37-s + 8·43-s − 3·47-s + 9·49-s + 3·53-s − 8·61-s + 6·65-s + 13·67-s − 2·73-s + 2·79-s − 18·83-s − 18·85-s + 3·89-s + 8·91-s − 6·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.51·7-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s − 1.43·31-s + 2.02·35-s + 0.328·37-s + 1.21·43-s − 0.437·47-s + 9/7·49-s + 0.412·53-s − 1.02·61-s + 0.744·65-s + 1.58·67-s − 0.234·73-s + 0.225·79-s − 1.97·83-s − 1.95·85-s + 0.317·89-s + 0.838·91-s − 0.615·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.38920658802096, −12.98519670233451, −12.37866939542061, −12.15570266576243, −11.92105066143001, −11.11558625590593, −10.74302593668604, −10.10364416122858, −9.639868206323314, −9.427298214500402, −8.668206688109040, −8.077281471416105, −7.750881554059574, −7.137540452115500, −6.922094295414320, −6.134216529199472, −5.683062671068904, −5.151640690824361, −4.349141838174842, −3.910992311825237, −3.420523625699933, −2.991060539688007, −2.436118641568799, −1.365420445513744, −0.6147876564320029, 0,
0.6147876564320029, 1.365420445513744, 2.436118641568799, 2.991060539688007, 3.420523625699933, 3.910992311825237, 4.349141838174842, 5.151640690824361, 5.683062671068904, 6.134216529199472, 6.922094295414320, 7.137540452115500, 7.750881554059574, 8.077281471416105, 8.668206688109040, 9.427298214500402, 9.639868206323314, 10.10364416122858, 10.74302593668604, 11.11558625590593, 11.92105066143001, 12.15570266576243, 12.37866939542061, 12.98519670233451, 13.38920658802096
