L(s) = 1 | + 0.947·2-s + 3.31·3-s − 1.10·4-s + 4.30·5-s + 3.13·6-s + 1.61·7-s − 2.93·8-s + 7.96·9-s + 4.07·10-s − 4.11·11-s − 3.64·12-s − 13-s + 1.52·14-s + 14.2·15-s − 0.584·16-s − 7.15·17-s + 7.54·18-s − 2.76·19-s − 4.73·20-s + 5.33·21-s − 3.89·22-s + 3.00·23-s − 9.73·24-s + 13.4·25-s − 0.947·26-s + 16.4·27-s − 1.77·28-s + ⋯ |
L(s) = 1 | + 0.670·2-s + 1.91·3-s − 0.550·4-s + 1.92·5-s + 1.28·6-s + 0.609·7-s − 1.03·8-s + 2.65·9-s + 1.28·10-s − 1.24·11-s − 1.05·12-s − 0.277·13-s + 0.408·14-s + 3.67·15-s − 0.146·16-s − 1.73·17-s + 1.77·18-s − 0.634·19-s − 1.05·20-s + 1.16·21-s − 0.831·22-s + 0.625·23-s − 1.98·24-s + 2.69·25-s − 0.185·26-s + 3.16·27-s − 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.918919343\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.918919343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 0.947T + 2T^{2} \) |
| 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 0.0391T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−9.310062711879285133925046267193, −8.917022687758469255493821241739, −8.346606034808234105229020102577, −7.20295587947982223417550656666, −6.26830420827083064958702203442, −5.03823597804513581368841872293, −4.65829867587977664777029549092, −3.29825502709479730554479620680, −2.41816904761179298179251535585, −1.86947297939714071022140257473,
1.86947297939714071022140257473, 2.41816904761179298179251535585, 3.29825502709479730554479620680, 4.65829867587977664777029549092, 5.03823597804513581368841872293, 6.26830420827083064958702203442, 7.20295587947982223417550656666, 8.346606034808234105229020102577, 8.917022687758469255493821241739, 9.310062711879285133925046267193