L(s) = 1 | − 0.291·2-s − 3.14·3-s − 1.91·4-s + 0.915·6-s + 1.20·7-s + 1.14·8-s + 6.86·9-s + 3.65·11-s + 6.01·12-s − 0.859·13-s − 0.351·14-s + 3.49·16-s − 6.72·17-s − 2·18-s − 1.51·19-s − 3.78·21-s − 1.06·22-s + 23-s − 3.58·24-s + 0.250·26-s − 12.1·27-s − 2.31·28-s + 0.548·29-s − 5.99·31-s − 3.30·32-s − 11.4·33-s + 1.95·34-s + ⋯ |
L(s) = 1 | − 0.206·2-s − 1.81·3-s − 0.957·4-s + 0.373·6-s + 0.456·7-s + 0.403·8-s + 2.28·9-s + 1.10·11-s + 1.73·12-s − 0.238·13-s − 0.0939·14-s + 0.874·16-s − 1.63·17-s − 0.471·18-s − 0.348·19-s − 0.826·21-s − 0.227·22-s + 0.208·23-s − 0.731·24-s + 0.0491·26-s − 2.33·27-s − 0.436·28-s + 0.101·29-s − 1.07·31-s − 0.583·32-s − 1.99·33-s + 0.335·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.291T + 2T^{2} \) |
| 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 + 0.859T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 29 | \( 1 - 0.548T + 29T^{2} \) |
| 31 | \( 1 + 5.99T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 - 0.0700T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 89 | \( 1 - 4.96T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−10.54455306696143694397595046816, −9.445568969775570819904226656462, −8.760693131504636087417321529469, −7.38735265934259576812546622562, −6.52881991139788445709684722705, −5.61280108071471714174037240422, −4.65808566254475230126659928981, −4.10772432859256061901774780325, −1.48363314772937410152673770352, 0,
1.48363314772937410152673770352, 4.10772432859256061901774780325, 4.65808566254475230126659928981, 5.61280108071471714174037240422, 6.52881991139788445709684722705, 7.38735265934259576812546622562, 8.760693131504636087417321529469, 9.445568969775570819904226656462, 10.54455306696143694397595046816