| L(s) = 1 | + 4·5-s − 3·9-s − 2·11-s − 2·13-s + 2·17-s − 2·19-s + 23-s + 11·25-s − 2·29-s + 4·37-s − 6·41-s − 10·43-s − 12·45-s + 4·53-s − 8·55-s + 12·59-s − 8·61-s − 8·65-s + 10·67-s − 6·73-s − 12·79-s + 9·81-s + 14·83-s + 8·85-s + 6·89-s − 8·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s − 0.371·29-s + 0.657·37-s − 0.937·41-s − 1.52·43-s − 1.78·45-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 − 1.35·79-s + 81-s + 1.53·83-s + 0.867·85-s + 0.635·89-s − 0.820·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 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.am |
| 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.g |
| 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.ag |
| 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
−14.41976191085705, −13.71174331689653, −13.43512406993471, −13.04583716465157, −12.43404815166447, −11.89563521503815, −11.28130808339425, −10.76879291847459, −10.11020510466776, −9.981721629630278, −9.339833151480264, −8.811151133668748, −8.353681684750622, −7.760713967919836, −6.969402849997744, −6.528113753470535, −5.942559537967180, −5.410452853584941, −5.187715442365930, −4.483559408373157, −3.448658319633598, −2.918385167064331, −2.296958658929959, −1.897989439012596, −0.9964623438255756, 0,
0.9964623438255756, 1.897989439012596, 2.296958658929959, 2.918385167064331, 3.448658319633598, 4.483559408373157, 5.187715442365930, 5.410452853584941, 5.942559537967180, 6.528113753470535, 6.969402849997744, 7.760713967919836, 8.353681684750622, 8.811151133668748, 9.339833151480264, 9.981721629630278, 10.11020510466776, 10.76879291847459, 11.28130808339425, 11.89563521503815, 12.43404815166447, 13.04583716465157, 13.43512406993471, 13.71174331689653, 14.41976191085705
