L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s − 0.517·5-s + (0.258 − 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (0.448 + 0.258i)10-s + (−0.707 + 0.707i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s − 0.517·5-s + (0.258 − 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (0.448 + 0.258i)10-s + (−0.707 + 0.707i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8214186410\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8214186410\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.517T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - 1.73iT - T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−9.082920873392461098713241720575, −8.352031788287855067036978996476, −7.67700868714691183083411916847, −6.98486926723948949878900151316, −5.88941615566807126915712466165, −4.98723108590272737712486745233, −3.79112932349929607899015162466, −3.60706404624817115904923309150, −2.54421220261743916796155680247, −1.26687336731633000569375645655,
0.71169294104140318309667719747, 1.67719399418523739410107668390, 2.84745919807383392357014496457, 3.71789778363650530790848110228, 5.10920250084778789919002868515, 6.00853929654260585757825571137, 6.37666567790245054109427685238, 7.49092553175390996201453501290, 7.72589702376111593609034524114, 8.496948870053982102717133709250