| L(s) = 1 | + 0.773·2-s + 3-s − 1.40·4-s − 1.00·5-s + 0.773·6-s − 0.385·7-s − 2.63·8-s + 9-s − 0.776·10-s + 3.88·11-s − 1.40·12-s − 3.10·13-s − 0.298·14-s − 1.00·15-s + 0.766·16-s + 0.0632·17-s + 0.773·18-s − 1.66·19-s + 1.40·20-s − 0.385·21-s + 3.00·22-s + 3.26·23-s − 2.63·24-s − 3.99·25-s − 2.40·26-s + 27-s + 0.540·28-s + ⋯ |
| L(s) = 1 | + 0.547·2-s + 0.577·3-s − 0.700·4-s − 0.449·5-s + 0.315·6-s − 0.145·7-s − 0.930·8-s + 0.333·9-s − 0.245·10-s + 1.17·11-s − 0.404·12-s − 0.860·13-s − 0.0798·14-s − 0.259·15-s + 0.191·16-s + 0.0153·17-s + 0.182·18-s − 0.382·19-s + 0.314·20-s − 0.0842·21-s + 0.640·22-s + 0.680·23-s − 0.537·24-s − 0.798·25-s − 0.470·26-s + 0.192·27-s + 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098741770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098741770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.773T + 2T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 0.385T + 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 - 0.0632T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 37 | \( 1 - 7.08T + 37T^{2} \) |
| 41 | \( 1 - 0.776T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.06T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 8.74T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.868800868487013399660148510519, −8.058020941232430947558191285214, −7.32266518097829385139725045082, −6.44428386806803009342682324669, −5.62915567725190998163682474023, −4.55000235326419375386583769334, −4.13064112123562327813452905354, −3.31275075408501061653610124300, −2.35513716498028736296394983853, −0.807581902029545871889591086377,
0.807581902029545871889591086377, 2.35513716498028736296394983853, 3.31275075408501061653610124300, 4.13064112123562327813452905354, 4.55000235326419375386583769334, 5.62915567725190998163682474023, 6.44428386806803009342682324669, 7.32266518097829385139725045082, 8.058020941232430947558191285214, 8.868800868487013399660148510519
