L(s) = 1 | − 2-s + 4-s − 5-s − 0.662·7-s − 8-s + 10-s − 2.66·11-s + 3.57·13-s + 0.662·14-s + 16-s + 3.18·17-s − 2.23·19-s − 20-s + 2.66·22-s − 7.42·23-s + 25-s − 3.57·26-s − 0.662·28-s + 3.15·29-s + 31-s − 32-s − 3.18·34-s + 0.662·35-s + 9.94·37-s + 2.23·38-s + 40-s + 3.15·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.250·7-s − 0.353·8-s + 0.316·10-s − 0.802·11-s + 0.991·13-s + 0.176·14-s + 0.250·16-s + 0.772·17-s − 0.513·19-s − 0.223·20-s + 0.567·22-s − 1.54·23-s + 0.200·25-s − 0.701·26-s − 0.125·28-s + 0.585·29-s + 0.179·31-s − 0.176·32-s − 0.545·34-s + 0.111·35-s + 1.63·37-s + 0.362·38-s + 0.158·40-s + 0.492·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9757756650\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9757756650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 0.662T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 - 9.83T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 1.61T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.634069265647341904124878412954, −8.037094310767888981675511806458, −7.64072452514144985987168052606, −6.44967253537457146027898926534, −6.03759429125677744679805746085, −4.92684752175251837113606763455, −3.88841394475031871037839270037, −3.06328348461637608444427343297, −1.98255781370053761107181410394, −0.67101706236641909982847929255,
0.67101706236641909982847929255, 1.98255781370053761107181410394, 3.06328348461637608444427343297, 3.88841394475031871037839270037, 4.92684752175251837113606763455, 6.03759429125677744679805746085, 6.44967253537457146027898926534, 7.64072452514144985987168052606, 8.037094310767888981675511806458, 8.634069265647341904124878412954