| L(s) = 1 | + 2-s − 3.03·3-s + 4-s − 3.03·6-s + 2.46·7-s + 8-s + 6.19·9-s + 0.728·11-s − 3.03·12-s − 6.23·13-s + 2.46·14-s + 16-s + 0.563·17-s + 6.19·18-s − 19-s − 7.49·21-s + 0.728·22-s + 4.63·23-s − 3.03·24-s − 6.23·26-s − 9.70·27-s + 2.46·28-s + 10.2·29-s + 6.06·31-s + 32-s − 2.21·33-s + 0.563·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s − 1.23·6-s + 0.933·7-s + 0.353·8-s + 2.06·9-s + 0.219·11-s − 0.875·12-s − 1.72·13-s + 0.660·14-s + 0.250·16-s + 0.136·17-s + 1.46·18-s − 0.229·19-s − 1.63·21-s + 0.155·22-s + 0.966·23-s − 0.619·24-s − 1.22·26-s − 1.86·27-s + 0.466·28-s + 1.89·29-s + 1.08·31-s + 0.176·32-s − 0.384·33-s + 0.0965·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.497478669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.497478669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 0.728T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 0.563T + 17T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 + 5.72T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 5.76T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.47812395760950593900892771064, −9.522129456197466936537817190141, −8.075894010530024812148858185685, −7.11204516473340676961782901409, −6.53049179238004494768702899980, −5.44986835051929002683014338050, −4.89242873982978485255593790771, −4.31480640408567750484737837572, −2.51140339553885340497874043869, −0.987570335650326101693547283034,
0.987570335650326101693547283034, 2.51140339553885340497874043869, 4.31480640408567750484737837572, 4.89242873982978485255593790771, 5.44986835051929002683014338050, 6.53049179238004494768702899980, 7.11204516473340676961782901409, 8.075894010530024812148858185685, 9.522129456197466936537817190141, 10.47812395760950593900892771064
