L(s) = 1 | + i·2-s + (0.608 − 0.793i)3-s − 4-s + (0.793 + 0.608i)6-s − i·8-s + (−0.258 − 0.965i)9-s + (−1.67 + 0.965i)11-s + (−0.608 + 0.793i)12-s + 16-s + (−0.793 + 1.37i)17-s + (0.965 − 0.258i)18-s + (−1.71 + 0.991i)19-s + (−0.965 − 1.67i)22-s + (−0.793 − 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + i·2-s + (0.608 − 0.793i)3-s − 4-s + (0.793 + 0.608i)6-s − i·8-s + (−0.258 − 0.965i)9-s + (−1.67 + 0.965i)11-s + (−0.608 + 0.793i)12-s + 16-s + (−0.793 + 1.37i)17-s + (0.965 − 0.258i)18-s + (−1.71 + 0.991i)19-s + (−0.965 − 1.67i)22-s + (−0.793 − 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3975812464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3975812464\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.608 + 0.793i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (1.71 - 0.991i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.608 - 1.05i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + 0.261T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.654953366909716700100530491120, −8.120001370011541191446529562573, −7.87400600140723664474375295160, −6.86890433829593232024792700384, −6.31416390544994401791798006948, −5.63371493534137269300714250882, −4.50536022004589332719336716744, −3.93558314191360134240929479443, −2.60989196730634652757558494254, −1.77092645283471072094192684338,
0.19504993542055297238883079821, 2.25014940926321746274229140812, 2.65228511845296838783266831210, 3.52978041854447262835585652930, 4.44629680210432670550395350323, 5.07664002857638193002513183811, 5.72842631071440392685702121115, 7.12394170221951934724960399562, 8.022175182867562349494555755146, 8.632638941790248035440952042793