L(s) = 1 | − 1.78·2-s + (−1.68 + 0.395i)3-s + 1.17·4-s − i·5-s + (3.00 − 0.704i)6-s − i·7-s + 1.47·8-s + (2.68 − 1.33i)9-s + 1.78i·10-s + (1.85 − 2.74i)11-s + (−1.98 + 0.464i)12-s − 5.68i·13-s + 1.78i·14-s + (0.395 + 1.68i)15-s − 4.96·16-s + 2.05·17-s + ⋯ |
L(s) = 1 | − 1.25·2-s + (−0.973 + 0.228i)3-s + 0.587·4-s − 0.447i·5-s + (1.22 − 0.287i)6-s − 0.377i·7-s + 0.519·8-s + (0.895 − 0.444i)9-s + 0.563i·10-s + (0.560 − 0.828i)11-s + (−0.571 + 0.134i)12-s − 1.57i·13-s + 0.476i·14-s + (0.102 + 0.435i)15-s − 1.24·16-s + 0.497·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5193182926\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5193182926\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.68 - 0.395i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-1.85 + 2.74i)T \) |
good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 13 | \( 1 + 5.68iT - 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 4.08iT - 19T^{2} \) |
| 23 | \( 1 - 5.58iT - 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 1.16iT - 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 4.49iT - 53T^{2} \) |
| 59 | \( 1 - 3.29iT - 59T^{2} \) |
| 61 | \( 1 + 2.93iT - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 7.13iT - 71T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 8.84iT - 79T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 + 0.807iT - 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−9.610598116507336246154785121932, −8.809568545334597325557516936877, −8.018427893247267051966053184743, −7.25071866620270606314133393503, −6.26878474226213886920622257077, −5.33502546807120925293900208681, −4.49639321220900139998563621538, −3.23575286723762747661565923476, −1.22464316035336197747397318980, −0.51934862862717031433487058493,
1.25424303114596663457575524572, 2.21986913199587006799261886965, 4.15660865513300648731494000460, 4.87178955720076249360582475068, 6.33088831407104428927643435424, 6.72260493270247550093370712726, 7.57487042085837094203577791926, 8.490784766472952355871476961706, 9.317898646176613996919212555179, 10.19757945891508275301011817226