| L(s) = 1 | − 2.33·2-s + 3.43·4-s − 3.08·5-s + 0.766·7-s − 3.35·8-s + 7.18·10-s − 3.16·11-s − 6.98·13-s − 1.78·14-s + 0.947·16-s − 4.31·17-s − 5.84·19-s − 10.6·20-s + 7.37·22-s + 0.909·23-s + 4.50·25-s + 16.2·26-s + 2.63·28-s − 5.86·29-s + 6.72·31-s + 4.50·32-s + 10.0·34-s − 2.36·35-s − 9.74·37-s + 13.6·38-s + 10.3·40-s + 10.8·41-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.71·4-s − 1.37·5-s + 0.289·7-s − 1.18·8-s + 2.27·10-s − 0.953·11-s − 1.93·13-s − 0.477·14-s + 0.236·16-s − 1.04·17-s − 1.34·19-s − 2.37·20-s + 1.57·22-s + 0.189·23-s + 0.900·25-s + 3.19·26-s + 0.498·28-s − 1.08·29-s + 1.20·31-s + 0.795·32-s + 1.72·34-s − 0.399·35-s − 1.60·37-s + 2.21·38-s + 1.63·40-s + 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1073575547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1073575547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 - 0.766T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 6.98T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 5.84T + 19T^{2} \) |
| 23 | \( 1 - 0.909T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 6.72T + 31T^{2} \) |
| 37 | \( 1 + 9.74T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 + 6.71T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 - 9.83T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 - 4.35T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−8.999024189159915595938216784501, −8.233575249956501406862322576793, −7.66733207255207529156390701583, −7.29313659581347059688997580392, −6.38121284058718770624843530567, −4.89931724684219822484272476385, −4.30490866517296221348159819736, −2.80256835783697008149243283459, −2.02721097196878560972268536282, −0.25580410885524105095536908903,
0.25580410885524105095536908903, 2.02721097196878560972268536282, 2.80256835783697008149243283459, 4.30490866517296221348159819736, 4.89931724684219822484272476385, 6.38121284058718770624843530567, 7.29313659581347059688997580392, 7.66733207255207529156390701583, 8.233575249956501406862322576793, 8.999024189159915595938216784501
