| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 6·13-s − 15-s + 8·17-s − 19-s − 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s − 2·31-s + 2·35-s + 2·37-s − 6·39-s − 12·41-s − 4·43-s − 45-s + 12·47-s − 3·49-s + 8·51-s − 10·53-s − 57-s − 6·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 1.94·17-s − 0.229·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.338·35-s + 0.328·37-s − 0.960·39-s − 1.87·41-s − 0.609·43-s − 0.149·45-s + 1.75·47-s − 3/7·49-s + 1.12·51-s − 1.37·53-s − 0.132·57-s − 0.781·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.443796329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443796329\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−15.76676936317416, −15.17181583463070, −14.60657572350145, −14.32244195880904, −13.67504555398813, −12.89999081447367, −12.49155210947518, −12.03644161681220, −11.55477316110137, −10.55013346911516, −9.975936845392362, −9.765976765634013, −9.108209500189870, −8.259052300920520, −7.827293992475887, −7.275587329253249, −6.748876513772151, −5.878651963229425, −5.228322417762556, −4.563609672043170, −3.652126177899483, −3.283352215403064, −2.506236760012082, −1.687558832109534, −0.4739773272584180,
0.4739773272584180, 1.687558832109534, 2.506236760012082, 3.283352215403064, 3.652126177899483, 4.563609672043170, 5.228322417762556, 5.878651963229425, 6.748876513772151, 7.275587329253249, 7.827293992475887, 8.259052300920520, 9.108209500189870, 9.765976765634013, 9.975936845392362, 10.55013346911516, 11.55477316110137, 12.03644161681220, 12.49155210947518, 12.89999081447367, 13.67504555398813, 14.32244195880904, 14.60657572350145, 15.17181583463070, 15.76676936317416
