| L(s) = 1 | + 2.62·2-s + 3-s + 4.86·4-s + 0.705·5-s + 2.62·6-s + 7-s + 7.51·8-s + 9-s + 1.84·10-s − 3.72·11-s + 4.86·12-s + 1.71·13-s + 2.62·14-s + 0.705·15-s + 9.96·16-s − 0.0523·17-s + 2.62·18-s − 1.01·19-s + 3.43·20-s + 21-s − 9.76·22-s + 5.73·23-s + 7.51·24-s − 4.50·25-s + 4.49·26-s + 27-s + 4.86·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.43·4-s + 0.315·5-s + 1.06·6-s + 0.377·7-s + 2.65·8-s + 0.333·9-s + 0.584·10-s − 1.12·11-s + 1.40·12-s + 0.475·13-s + 0.700·14-s + 0.182·15-s + 2.49·16-s − 0.0126·17-s + 0.617·18-s − 0.233·19-s + 0.768·20-s + 0.218·21-s − 2.08·22-s + 1.19·23-s + 1.53·24-s − 0.900·25-s + 0.880·26-s + 0.192·27-s + 0.920·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.626490922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.626490922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 5 | \( 1 - 0.705T + 5T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 + 0.0523T + 17T^{2} \) |
| 19 | \( 1 + 1.01T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 + 8.95T + 43T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 4.04T + 59T^{2} \) |
| 61 | \( 1 - 9.56T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−7.997263646103293720643669260796, −7.74242901482020184670522458105, −6.62458878862847344058809724981, −6.15631057962714538743396972912, −5.19411524038246600936125277714, −4.75292327659687991018717718120, −3.91057527308884561826560561691, −2.98670666117461810485762394865, −2.49109500939797034433245876114, −1.45820317937687475956302889022,
1.45820317937687475956302889022, 2.49109500939797034433245876114, 2.98670666117461810485762394865, 3.91057527308884561826560561691, 4.75292327659687991018717718120, 5.19411524038246600936125277714, 6.15631057962714538743396972912, 6.62458878862847344058809724981, 7.74242901482020184670522458105, 7.997263646103293720643669260796
