L(s) = 1 | + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (−0.326 + 0.118i)22-s + (−0.326 + 1.85i)24-s + (−0.939 − 0.342i)25-s + (1.43 − 2.49i)27-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (−0.326 + 0.118i)22-s + (−0.326 + 1.85i)24-s + (−0.939 − 0.342i)25-s + (1.43 − 2.49i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.238179342\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.238179342\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
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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
−8.817752783677320103136559892878, −7.985611599063441443597862973735, −7.67118580597858557813780245629, −6.80611959826349112229054261516, −6.30335289047551213423021650379, −4.97747714664260495981443775951, −4.19615159672880746910961112043, −3.35999494955568679316791089138, −2.62594011442211910364041881227, −1.82203085985544451982701979060,
1.77356914607159640959250857582, 2.38864530148281236270064334462, 3.42729402701462546851841921927, 3.82131075727552852412857379963, 4.67870156052823582254878464448, 5.50050163419007950546888852269, 6.65039080449053016312132158146, 7.46956369552738232636715444644, 8.488062942817625292891750516113, 8.818534808166803290555612683672