L(s) = 1 | + 2-s + 0.510·3-s + 4-s − 1.22·5-s + 0.510·6-s + 7-s + 8-s − 2.73·9-s − 1.22·10-s + 5.54·11-s + 0.510·12-s + 5.10·13-s + 14-s − 0.627·15-s + 16-s + 4.15·17-s − 2.73·18-s + 5.09·19-s − 1.22·20-s + 0.510·21-s + 5.54·22-s − 6.28·23-s + 0.510·24-s − 3.48·25-s + 5.10·26-s − 2.93·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.294·3-s + 0.5·4-s − 0.549·5-s + 0.208·6-s + 0.377·7-s + 0.353·8-s − 0.913·9-s − 0.388·10-s + 1.67·11-s + 0.147·12-s + 1.41·13-s + 0.267·14-s − 0.162·15-s + 0.250·16-s + 1.00·17-s − 0.645·18-s + 1.16·19-s − 0.274·20-s + 0.111·21-s + 1.18·22-s − 1.31·23-s + 0.104·24-s − 0.697·25-s + 1.00·26-s − 0.563·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.947133027\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.947133027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 0.510T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + 0.696T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.0901T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 1.18T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 4.64T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 + 8.26T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 + 6.00T + 83T^{2} \) |
| 89 | \( 1 - 0.670T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.941260955229042783983825306834, −7.51318183782671429035055173676, −6.42711824571341968410155630944, −5.92141867641321926958477537267, −5.32702052129479463794066753685, −4.04325645012233524948014501662, −3.81419271379029377484033022824, −3.11001154131569794478474147269, −1.88144043710396511883004794059, −0.990202613345958798190863640820,
0.990202613345958798190863640820, 1.88144043710396511883004794059, 3.11001154131569794478474147269, 3.81419271379029377484033022824, 4.04325645012233524948014501662, 5.32702052129479463794066753685, 5.92141867641321926958477537267, 6.42711824571341968410155630944, 7.51318183782671429035055173676, 7.941260955229042783983825306834