L(s) = 1 | + (−0.885 + 0.464i)2-s + (0.568 − 0.822i)4-s + (0.120 + 0.992i)5-s + (−0.120 + 0.992i)8-s + (−0.885 + 0.464i)9-s + (−0.568 − 0.822i)10-s + (0.120 + 0.992i)13-s + (−0.354 − 0.935i)16-s + (1.63 + 0.198i)17-s + (0.568 − 0.822i)18-s + (0.885 + 0.464i)20-s + (−0.970 + 0.239i)25-s + (−0.568 − 0.822i)26-s + (1.71 − 0.902i)29-s + (0.748 + 0.663i)32-s + ⋯ |
L(s) = 1 | + (−0.885 + 0.464i)2-s + (0.568 − 0.822i)4-s + (0.120 + 0.992i)5-s + (−0.120 + 0.992i)8-s + (−0.885 + 0.464i)9-s + (−0.568 − 0.822i)10-s + (0.120 + 0.992i)13-s + (−0.354 − 0.935i)16-s + (1.63 + 0.198i)17-s + (0.568 − 0.822i)18-s + (0.885 + 0.464i)20-s + (−0.970 + 0.239i)25-s + (−0.568 − 0.822i)26-s + (1.71 − 0.902i)29-s + (0.748 + 0.663i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7301445642\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7301445642\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.885 - 0.464i)T \) |
| 5 | \( 1 + (-0.120 - 0.992i)T \) |
| 13 | \( 1 + (-0.120 - 0.992i)T \) |
good | 3 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 7 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 11 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 0.198i)T + (0.970 + 0.239i)T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-1.71 + 0.902i)T + (0.568 - 0.822i)T^{2} \) |
| 31 | \( 1 + (0.120 - 0.992i)T^{2} \) |
| 37 | \( 1 + (0.530 - 0.470i)T + (0.120 - 0.992i)T^{2} \) |
| 41 | \( 1 + (-0.222 - 0.902i)T + (-0.885 + 0.464i)T^{2} \) |
| 43 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 47 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 53 | \( 1 + (1.97 + 0.239i)T + (0.970 + 0.239i)T^{2} \) |
| 59 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 61 | \( 1 + (-0.234 - 1.92i)T + (-0.970 + 0.239i)T^{2} \) |
| 67 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 71 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 73 | \( 1 + (1.77 + 0.929i)T + (0.568 + 0.822i)T^{2} \) |
| 79 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 83 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 89 | \( 1 - 1.87iT - T^{2} \) |
| 97 | \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \) |
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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
−9.030683431278842116147874906434, −8.071127358582288190957926504372, −7.82816884882445594462173379312, −6.77851156133101562225662800619, −6.27865417524294233327767382443, −5.59665830194541703920188073353, −4.63716970028917691264503013545, −3.25401182411022236105902592735, −2.54912615741238929960698502857, −1.44415148252525184296588706316,
0.61467770296203168875786660298, 1.58484902174047982122282055118, 2.98191512723398991450051504129, 3.41975457919605500056803012520, 4.71222765741094858105164633593, 5.57947473562830537586728842957, 6.25590767740215025788825409349, 7.35153730543126004803441680432, 8.107130582996127864660738426574, 8.516456223271239767464584039001