L(s) = 1 | + (0.955 + 0.294i)4-s + (0.142 + 1.90i)5-s + (0.0747 + 0.997i)7-s + (0.365 − 0.930i)9-s + (−0.722 − 1.84i)11-s + (0.826 + 0.563i)16-s + (0.326 + 0.302i)17-s + (−0.5 − 0.866i)19-s + (−0.425 + 1.86i)20-s + (−0.535 + 0.496i)23-s + (−2.62 + 0.395i)25-s + (−0.222 + 0.974i)28-s + (−1.88 + 0.284i)35-s + (0.623 − 0.781i)36-s + (1.32 − 0.636i)43-s + (−0.147 − 1.97i)44-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)4-s + (0.142 + 1.90i)5-s + (0.0747 + 0.997i)7-s + (0.365 − 0.930i)9-s + (−0.722 − 1.84i)11-s + (0.826 + 0.563i)16-s + (0.326 + 0.302i)17-s + (−0.5 − 0.866i)19-s + (−0.425 + 1.86i)20-s + (−0.535 + 0.496i)23-s + (−2.62 + 0.395i)25-s + (−0.222 + 0.974i)28-s + (−1.88 + 0.284i)35-s + (0.623 − 0.781i)36-s + (1.32 − 0.636i)43-s + (−0.147 − 1.97i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.244847265\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244847265\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.0747 - 0.997i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 3 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 5 | \( 1 + (-0.142 - 1.90i)T + (-0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.326 - 0.302i)T + (0.0747 + 0.997i)T^{2} \) |
| 23 | \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-1.32 + 0.636i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (1.63 + 0.246i)T + (0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 61 | \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.988 + 0.149i)T + (0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.914 + 1.14i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62074879254784528359023380258, −9.780251030726493296002586335987, −8.613686116832009583081906777426, −7.77788784388352566434682172579, −6.86263496252385146540491260706, −6.17333753764184944769059383947, −5.72208223500396974406671604544, −3.56510916960175904583355203195, −3.05578402586603668746804653317, −2.20921680240799776125907303548,
1.41498514178854086254503295525, 2.18147832742595872276909544889, 4.20550379886040510390486990879, 4.80143418963005399798868329223, 5.58711090708542263541373768980, 6.86408239301547854348197130832, 7.80991362831868491878477835764, 8.077315066411742335466464989295, 9.640550434923029321259604974228, 10.00998974299057213813340430827