| L(s) = 1 | + 1.93·2-s − 3-s + 1.73·4-s − 1.93·6-s − 0.517·8-s + 9-s − 3.46·11-s − 1.73·12-s + 4·13-s − 4.46·16-s + 4·17-s + 1.93·18-s + 0.378·19-s − 6.69·22-s − 6.31·23-s + 0.517·24-s + 7.72·26-s − 27-s + 8.92·29-s + 7.34·31-s − 7.58·32-s + 3.46·33-s + 7.72·34-s + 1.73·36-s + 0.757·37-s + 0.732·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 0.577·3-s + 0.866·4-s − 0.788·6-s − 0.183·8-s + 0.333·9-s − 1.04·11-s − 0.500·12-s + 1.10·13-s − 1.11·16-s + 0.970·17-s + 0.455·18-s + 0.0869·19-s − 1.42·22-s − 1.31·23-s + 0.105·24-s + 1.51·26-s − 0.192·27-s + 1.65·29-s + 1.31·31-s − 1.34·32-s + 0.603·33-s + 1.32·34-s + 0.288·36-s + 0.124·37-s + 0.118·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.014868192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.014868192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 0.378T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 0.757T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−8.286788421652170446003523135259, −7.75466509626511695250914348622, −6.61971251072583585949284566321, −6.05248247619972424478048085671, −5.53641517608998441110701142752, −4.73493628508467850837772194489, −4.07572911158261936234386986123, −3.20201120635432632656029406049, −2.36238977081349655492779706471, −0.854637472226835135908143334879,
0.854637472226835135908143334879, 2.36238977081349655492779706471, 3.20201120635432632656029406049, 4.07572911158261936234386986123, 4.73493628508467850837772194489, 5.53641517608998441110701142752, 6.05248247619972424478048085671, 6.61971251072583585949284566321, 7.75466509626511695250914348622, 8.286788421652170446003523135259
