| L(s) = 1 | − 4·5-s − 4·7-s + 2·11-s + 2·13-s − 2·17-s + 2·19-s + 11·25-s + 2·29-s + 16·35-s − 4·37-s − 6·41-s − 10·43-s + 9·49-s + 4·53-s − 8·55-s − 12·59-s − 8·61-s − 8·65-s + 10·67-s + 6·73-s − 8·77-s − 12·79-s + 14·83-s + 8·85-s − 6·89-s − 8·91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 0.458·19-s + 11/5·25-s + 0.371·29-s + 2.70·35-s − 0.657·37-s − 0.937·41-s − 1.52·43-s + 9/7·49-s + 0.549·53-s − 1.07·55-s − 1.56·59-s − 1.02·61-s − 0.992·65-s + 1.22·67-s + 0.702·73-s − 0.911·77-s − 1.35·79-s + 1.53·83-s + 0.867·85-s − 0.635·89-s − 0.838·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.06040545348051, −12.56006423489695, −12.11658516117853, −11.86847637466204, −11.43540231675856, −10.81221202462382, −10.53988082276707, −9.904167192842598, −9.366357253335238, −9.009166544321019, −8.469371994106643, −8.074070274136916, −7.596617135401983, −6.852209869367162, −6.744498347692074, −6.375792742823434, −5.555899834095623, −5.016589544920038, −4.380926554117838, −3.891522016841451, −3.534255739924854, −3.131271280712434, −2.639881412127935, −1.590603678297354, −0.9790862148873024, 0, 0,
0.9790862148873024, 1.590603678297354, 2.639881412127935, 3.131271280712434, 3.534255739924854, 3.891522016841451, 4.380926554117838, 5.016589544920038, 5.555899834095623, 6.375792742823434, 6.744498347692074, 6.852209869367162, 7.596617135401983, 8.074070274136916, 8.469371994106643, 9.009166544321019, 9.366357253335238, 9.904167192842598, 10.53988082276707, 10.81221202462382, 11.43540231675856, 11.86847637466204, 12.11658516117853, 12.56006423489695, 13.06040545348051
