| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 0.666·5-s + 0.549·7-s + 4.39·8-s − 1.60·10-s + 0.262·11-s + 3.05·13-s + 1.32·14-s + 2.96·16-s + 2.00·17-s + 1.22·19-s − 2.54·20-s + 0.633·22-s + 6.88·23-s − 4.55·25-s + 7.36·26-s + 2.10·28-s + 5.64·29-s + 2.28·31-s − 1.63·32-s + 4.84·34-s − 0.366·35-s + 8.46·37-s + 2.95·38-s − 2.93·40-s + 1.90·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.298·5-s + 0.207·7-s + 1.55·8-s − 0.508·10-s + 0.0791·11-s + 0.846·13-s + 0.354·14-s + 0.741·16-s + 0.487·17-s + 0.280·19-s − 0.569·20-s + 0.135·22-s + 1.43·23-s − 0.911·25-s + 1.44·26-s + 0.397·28-s + 1.04·29-s + 0.409·31-s − 0.289·32-s + 0.831·34-s − 0.0619·35-s + 1.39·37-s + 0.478·38-s − 0.463·40-s + 0.298·41-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
| \(L(1)\) |
\(\approx\) |
\(5.144412925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144412925\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 + 0.666T + 5T^{2} \) |
| 7 | \( 1 - 0.549T + 7T^{2} \) |
| 11 | \( 1 - 0.262T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 - 6.88T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 - 8.46T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 4.78T + 71T^{2} \) |
| 73 | \( 1 - 2.16T + 73T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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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
−8.998912483737492355714702133502, −8.087917963160308646622567659082, −7.27625631604733985161778151503, −6.44084387718289344379077916707, −5.80714872734607554463143139331, −4.94631109662104000717938274757, −4.27087225045902185797700238165, −3.40822117222377142217690797537, −2.69640561942836688736656563524, −1.31132000116664843546780904914,
1.31132000116664843546780904914, 2.69640561942836688736656563524, 3.40822117222377142217690797537, 4.27087225045902185797700238165, 4.94631109662104000717938274757, 5.80714872734607554463143139331, 6.44084387718289344379077916707, 7.27625631604733985161778151503, 8.087917963160308646622567659082, 8.998912483737492355714702133502
