L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 0.879·18-s + (−1.43 − 0.524i)22-s + (0.0603 + 0.342i)24-s + (−0.939 + 0.342i)25-s + (0.326 + 0.565i)27-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 0.879·18-s + (−1.43 − 0.524i)22-s + (0.0603 + 0.342i)24-s + (−0.939 + 0.342i)25-s + (0.326 + 0.565i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.028790105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028790105\) |
\(L(1)\) |
|
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 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 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 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
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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
−8.717485552745473278819404447412, −7.85854566706406299305227781977, −6.73652490669601053201376145462, −5.95082179180660964978290334059, −5.65577245711604099628777766723, −4.63864885002543198334428166261, −3.64229522992081764219625383405, −3.00591448165143880711705477898, −1.94653383442201932072962079994, −0.47606307668568991245867624131,
2.22434237898600641808525103009, 2.80241878517462114835139555339, 4.21770995754293184603222313351, 4.67416843674059025545600575245, 5.50616176071731815121810620539, 6.10527069204384439148311316337, 7.12188569879811014300278436540, 7.60375208343662713417682518666, 8.346666915341636495402338832131, 9.206984806885854277044666080505