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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09385415510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09385415510\) |
\(L(1)\) |
|
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
−9.300578740683266174215555312500, −8.288041037544156503403387077349, −7.61282428966312277608129330814, −6.92036409895903882398410119570, −6.26062814025734981221790149030, −5.74386165215762020616071346602, −5.05044401495856997240351112072, −4.15297483425569118630151391726, −2.19021535018333059278366732328, −1.22378975267257856609984843448,
0.10933718116974365432406856037, 1.46312046178725462402751610280, 2.82046631317326504938761186660, 4.14780856139790101306455999087, 4.50865633155692158199332380516, 5.57091522680854053958043657039, 6.34662690038194586374154317723, 7.08996434169946346857218403597, 7.81951062968417567504330581476, 9.007120485114646616896224879420