| L(s) = 1 | + 1.20·2-s + 3-s − 0.540·4-s + 2.72·5-s + 1.20·6-s − 1.05·7-s − 3.06·8-s + 9-s + 3.29·10-s + 0.0950·11-s − 0.540·12-s + 1.29·13-s − 1.27·14-s + 2.72·15-s − 2.62·16-s + 6.72·17-s + 1.20·18-s + 1.49·19-s − 1.47·20-s − 1.05·21-s + 0.114·22-s + 23-s − 3.06·24-s + 2.45·25-s + 1.56·26-s + 27-s + 0.569·28-s + ⋯ |
| L(s) = 1 | + 0.854·2-s + 0.577·3-s − 0.270·4-s + 1.22·5-s + 0.493·6-s − 0.397·7-s − 1.08·8-s + 0.333·9-s + 1.04·10-s + 0.0286·11-s − 0.156·12-s + 0.359·13-s − 0.339·14-s + 0.704·15-s − 0.656·16-s + 1.63·17-s + 0.284·18-s + 0.341·19-s − 0.330·20-s − 0.229·21-s + 0.0244·22-s + 0.208·23-s − 0.626·24-s + 0.490·25-s + 0.307·26-s + 0.192·27-s + 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.613002126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.613002126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.0950T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 - 5.28T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 0.852T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 - 7.14T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 9.23T + 97T^{2} \) |
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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
−9.226689244020388014299593685989, −8.538962257333366111688814926613, −7.57755293248374410643884200817, −6.50896502500409671254559081702, −5.79414318413880455357805695107, −5.25634651970317936830351408192, −4.17680364030518623692746728495, −3.28617593625171950373899900416, −2.57997671719373602020978670277, −1.19825533744524992424105785928,
1.19825533744524992424105785928, 2.57997671719373602020978670277, 3.28617593625171950373899900416, 4.17680364030518623692746728495, 5.25634651970317936830351408192, 5.79414318413880455357805695107, 6.50896502500409671254559081702, 7.57755293248374410643884200817, 8.538962257333366111688814926613, 9.226689244020388014299593685989
