| L(s) = 1 | + 2·5-s + 4·11-s + 13-s + 2·17-s − 4·19-s − 4·23-s − 25-s − 2·29-s − 2·37-s − 2·41-s + 4·43-s + 8·47-s − 2·53-s + 8·55-s − 4·59-s + 14·61-s + 2·65-s + 12·67-s + 8·71-s − 14·73-s + 12·79-s + 4·83-s + 4·85-s + 14·89-s − 8·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s + 1.07·55-s − 0.520·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.35·79-s + 0.439·83-s + 0.433·85-s + 1.48·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612488983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612488983\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 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.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−12.41136885506924, −12.09200278061214, −11.69854550316894, −11.10431841610812, −10.62086844909763, −10.30889776316010, −9.624902043082020, −9.442028037961751, −8.983727445978225, −8.419890160548981, −8.000916802921942, −7.475183731988373, −6.729861002845089, −6.517487683102745, −6.082074272405170, −5.421118130433535, −5.286306654355560, −4.283406778314211, −4.002063300495001, −3.587840809551341, −2.790707101613055, −2.160450451755396, −1.809556735634757, −1.174855165543625, −0.5007553511630635,
0.5007553511630635, 1.174855165543625, 1.809556735634757, 2.160450451755396, 2.790707101613055, 3.587840809551341, 4.002063300495001, 4.283406778314211, 5.286306654355560, 5.421118130433535, 6.082074272405170, 6.517487683102745, 6.729861002845089, 7.475183731988373, 8.000916802921942, 8.419890160548981, 8.983727445978225, 9.442028037961751, 9.624902043082020, 10.30889776316010, 10.62086844909763, 11.10431841610812, 11.69854550316894, 12.09200278061214, 12.41136885506924
