L(s) = 1 | + 0.363·2-s − 3-s − 1.86·4-s − 5-s − 0.363·6-s + 3.29·7-s − 1.40·8-s + 9-s − 0.363·10-s + 5.11·11-s + 1.86·12-s + 3.88·13-s + 1.19·14-s + 15-s + 3.22·16-s + 1.77·17-s + 0.363·18-s + 8.58·19-s + 1.86·20-s − 3.29·21-s + 1.85·22-s + 3.04·23-s + 1.40·24-s + 25-s + 1.41·26-s − 27-s − 6.15·28-s + ⋯ |
L(s) = 1 | + 0.256·2-s − 0.577·3-s − 0.933·4-s − 0.447·5-s − 0.148·6-s + 1.24·7-s − 0.496·8-s + 0.333·9-s − 0.114·10-s + 1.54·11-s + 0.539·12-s + 1.07·13-s + 0.319·14-s + 0.258·15-s + 0.806·16-s + 0.429·17-s + 0.0856·18-s + 1.96·19-s + 0.417·20-s − 0.718·21-s + 0.395·22-s + 0.634·23-s + 0.286·24-s + 0.200·25-s + 0.276·26-s − 0.192·27-s − 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.223123806\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.223123806\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 0.363T + 2T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 - 3.88T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 - 8.58T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 - 0.998T + 29T^{2} \) |
| 31 | \( 1 + 0.261T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 0.494T + 41T^{2} \) |
| 43 | \( 1 + 0.551T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 0.336T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.26T + 83T^{2} \) |
| 89 | \( 1 + 5.55T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
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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
−8.103019944975080981091712743639, −7.40579594534792522495103763251, −6.63363985786989869980815535905, −5.59967990613397304503609728210, −5.32833854900630305353465145657, −4.32351216926089775654584691656, −3.95932806524670816937897977256, −3.09478701563262596193439456128, −1.30431137206631108118971735843, −0.997613279315124446937701125203,
0.997613279315124446937701125203, 1.30431137206631108118971735843, 3.09478701563262596193439456128, 3.95932806524670816937897977256, 4.32351216926089775654584691656, 5.32833854900630305353465145657, 5.59967990613397304503609728210, 6.63363985786989869980815535905, 7.40579594534792522495103763251, 8.103019944975080981091712743639