| L(s) = 1 | − 5-s − 7-s − 6·11-s + 2·13-s + 6·17-s − 19-s − 6·23-s + 25-s − 6·29-s − 8·31-s + 35-s + 2·37-s + 12·41-s − 8·43-s + 12·47-s + 49-s − 6·53-s + 6·55-s − 6·59-s − 4·61-s − 2·65-s + 10·67-s − 16·73-s + 6·77-s + 10·79-s + 12·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s + 1.45·17-s − 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s + 1.87·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.809·55-s − 0.781·59-s − 0.512·61-s − 0.248·65-s + 1.22·67-s − 1.87·73-s + 0.683·77-s + 1.12·79-s + 1.31·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6728759631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6728759631\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 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
−13.82713890247524, −13.17294211467441, −12.79644962240304, −12.47625545980489, −11.89317151728986, −11.29724308827789, −10.68955386347475, −10.52165144872556, −9.888128777388064, −9.328592822962284, −8.878109665645293, −8.018697421995004, −7.736418259905783, −7.575213172392887, −6.731732851310323, −5.932342055809812, −5.675279419584836, −5.186913453036975, −4.405330912462272, −3.756983655170824, −3.375043245913462, −2.645397566229408, −2.087226837940655, −1.212666168194736, −0.2682406501035470,
0.2682406501035470, 1.212666168194736, 2.087226837940655, 2.645397566229408, 3.375043245913462, 3.756983655170824, 4.405330912462272, 5.186913453036975, 5.675279419584836, 5.932342055809812, 6.731732851310323, 7.575213172392887, 7.736418259905783, 8.018697421995004, 8.878109665645293, 9.328592822962284, 9.888128777388064, 10.52165144872556, 10.68955386347475, 11.29724308827789, 11.89317151728986, 12.47625545980489, 12.79644962240304, 13.17294211467441, 13.82713890247524
