L(s) = 1 | + (1.59 − 0.580i)2-s + (−0.130 + 0.737i)3-s + (1.43 − 1.20i)4-s + (0.220 + 1.25i)6-s + (−0.241 − 0.419i)7-s + (0.743 − 1.28i)8-s + (0.412 + 0.150i)9-s + (0.5 − 0.866i)11-s + (0.703 + 1.21i)12-s + (−0.173 − 0.984i)13-s + (−0.628 − 0.527i)14-s + (0.112 − 0.635i)16-s + 0.744·18-s + (−0.990 − 0.139i)19-s + (0.340 − 0.123i)21-s + (0.294 − 1.67i)22-s + ⋯ |
L(s) = 1 | + (1.59 − 0.580i)2-s + (−0.130 + 0.737i)3-s + (1.43 − 1.20i)4-s + (0.220 + 1.25i)6-s + (−0.241 − 0.419i)7-s + (0.743 − 1.28i)8-s + (0.412 + 0.150i)9-s + (0.5 − 0.866i)11-s + (0.703 + 1.21i)12-s + (−0.173 − 0.984i)13-s + (−0.628 − 0.527i)14-s + (0.112 − 0.635i)16-s + 0.744·18-s + (−0.990 − 0.139i)19-s + (0.340 − 0.123i)21-s + (0.294 − 1.67i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.949531115\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.949531115\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.990 + 0.139i)T \) |
good | 2 | \( 1 + (-1.59 + 0.580i)T + (0.766 - 0.642i)T^{2} \) |
| 3 | \( 1 + (0.130 - 0.737i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.241 + 0.419i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.856 + 0.718i)T + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.333 - 1.89i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.943 - 0.791i)T + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.997 + 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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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
−9.128389549944718722830401925677, −8.192042621748345247764649632940, −7.06475335200958366692826510940, −6.35487408531034979334260300346, −5.55326740058384434515273035070, −4.82142779517673540643808227862, −4.21120377712706302969977888549, −3.41460510533341057161225696052, −2.80188806670403154914115405048, −1.37367954432414737424597964922,
1.74463996976702558433706954706, 2.56790361375951040212264020786, 3.84407886372942230040369583415, 4.35729584954150082741723123013, 5.21497382401704531634131670540, 6.12779321339511502195038691145, 6.77945517820325524003980761874, 7.03090112816554610037496187387, 7.934468544922990492674892672972, 9.015264999938226617937443078894