| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 2·11-s + 4·13-s + 15-s − 6·17-s + 19-s − 2·21-s − 4·23-s + 25-s + 27-s + 8·29-s − 8·31-s + 2·33-s − 2·35-s + 8·37-s + 4·39-s − 2·43-s + 45-s + 4·47-s − 3·49-s − 6·51-s − 6·53-s + 2·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.229·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.348·33-s − 0.338·35-s + 1.31·37-s + 0.640·39-s − 0.304·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s + 0.132·57-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
| \(L(1)\) |
\(\approx\) |
\(2.873636425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.873636425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−15.92081586716718, −15.23286695693904, −14.64395425785689, −14.05398099350469, −13.60842255290848, −13.08014995012537, −12.74040332682879, −11.90004985960668, −11.33731702986418, −10.68495345025525, −10.16957448906200, −9.315577508035043, −9.235405824161170, −8.468607322246047, −7.964080552135377, −7.048462053327610, −6.435424648346148, −6.215170202229824, −5.299030703084599, −4.387648249044611, −3.864248737406804, −3.189879892061435, −2.388465844437560, −1.699748109998054, −0.6915184574415910,
0.6915184574415910, 1.699748109998054, 2.388465844437560, 3.189879892061435, 3.864248737406804, 4.387648249044611, 5.299030703084599, 6.215170202229824, 6.435424648346148, 7.048462053327610, 7.964080552135377, 8.468607322246047, 9.235405824161170, 9.315577508035043, 10.16957448906200, 10.68495345025525, 11.33731702986418, 11.90004985960668, 12.74040332682879, 13.08014995012537, 13.60842255290848, 14.05398099350469, 14.64395425785689, 15.23286695693904, 15.92081586716718
