| L(s) = 1 | + 0.188·2-s − 3-s − 1.96·4-s − 0.188·6-s − 1.64·7-s − 0.745·8-s + 9-s + 1.96·12-s + 6.71·13-s − 0.310·14-s + 3.78·16-s + 1.16·17-s + 0.188·18-s + 2.24·19-s + 1.64·21-s + 8.49·23-s + 0.745·24-s + 1.26·26-s − 27-s + 3.23·28-s + 9.45·29-s + 3.73·31-s + 2.20·32-s + 0.219·34-s − 1.96·36-s + 9.66·37-s + 0.421·38-s + ⋯ |
| L(s) = 1 | + 0.132·2-s − 0.577·3-s − 0.982·4-s − 0.0767·6-s − 0.623·7-s − 0.263·8-s + 0.333·9-s + 0.567·12-s + 1.86·13-s − 0.0828·14-s + 0.947·16-s + 0.283·17-s + 0.0443·18-s + 0.514·19-s + 0.359·21-s + 1.77·23-s + 0.152·24-s + 0.247·26-s − 0.192·27-s + 0.612·28-s + 1.75·29-s + 0.670·31-s + 0.389·32-s + 0.0376·34-s − 0.327·36-s + 1.58·37-s + 0.0684·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.592154948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592154948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.188T + 2T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 13 | \( 1 - 6.71T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 - 9.45T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 + 6.80T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 - 7.76T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 5.86T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−7.87632390288884794992962242642, −6.75174274940702736961011335160, −6.35886288959618713685857407286, −5.66941835682503377914292473551, −4.91226523481200646652910727871, −4.37675063272483365476503341018, −3.38569761297906804331275577649, −3.04346808663754793179164050897, −1.29556318619734457063702552717, −0.73246901646390740203509133931,
0.73246901646390740203509133931, 1.29556318619734457063702552717, 3.04346808663754793179164050897, 3.38569761297906804331275577649, 4.37675063272483365476503341018, 4.91226523481200646652910727871, 5.66941835682503377914292473551, 6.35886288959618713685857407286, 6.75174274940702736961011335160, 7.87632390288884794992962242642
