| L(s) = 1 | + 2-s − 1.13·3-s + 4-s + 2.43·5-s − 1.13·6-s − 3.63·7-s + 8-s − 1.71·9-s + 2.43·10-s + 2.43·11-s − 1.13·12-s − 3.63·14-s − 2.75·15-s + 16-s + 0.797·17-s − 1.71·18-s − 19-s + 2.43·20-s + 4.11·21-s + 2.43·22-s + 1.81·23-s − 1.13·24-s + 0.905·25-s + 5.34·27-s − 3.63·28-s + 0.247·29-s − 2.75·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.654·3-s + 0.5·4-s + 1.08·5-s − 0.462·6-s − 1.37·7-s + 0.353·8-s − 0.571·9-s + 0.768·10-s + 0.732·11-s − 0.327·12-s − 0.970·14-s − 0.711·15-s + 0.250·16-s + 0.193·17-s − 0.404·18-s − 0.229·19-s + 0.543·20-s + 0.898·21-s + 0.518·22-s + 0.377·23-s − 0.231·24-s + 0.181·25-s + 1.02·27-s − 0.686·28-s + 0.0459·29-s − 0.502·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.474033722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474033722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 17 | \( 1 - 0.797T + 17T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 - 0.247T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 - 0.823T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 6.56T + 73T^{2} \) |
| 79 | \( 1 - 5.06T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.908573305578866024445359682603, −6.75943081712332295827688804810, −6.46348203108341511815681120601, −5.96959969938214862025160575265, −5.34500582193783425569940787725, −4.54980916404930958424018182408, −3.51360325452017685807011722919, −2.90448787097101219185946296021, −1.99028670707954886736538799059, −0.74365034283077501686520179317,
0.74365034283077501686520179317, 1.99028670707954886736538799059, 2.90448787097101219185946296021, 3.51360325452017685807011722919, 4.54980916404930958424018182408, 5.34500582193783425569940787725, 5.96959969938214862025160575265, 6.46348203108341511815681120601, 6.75943081712332295827688804810, 7.908573305578866024445359682603
