| L(s) = 1 | − 1.22·2-s + 3-s − 0.502·4-s + 1.43·5-s − 1.22·6-s − 1.77·7-s + 3.06·8-s + 9-s − 1.76·10-s − 4.03·11-s − 0.502·12-s + 1.49·13-s + 2.17·14-s + 1.43·15-s − 2.74·16-s + 17-s − 1.22·18-s + 0.978·19-s − 0.724·20-s − 1.77·21-s + 4.93·22-s + 2.98·23-s + 3.06·24-s − 2.92·25-s − 1.82·26-s + 27-s + 0.894·28-s + ⋯ |
| L(s) = 1 | − 0.865·2-s + 0.577·3-s − 0.251·4-s + 0.643·5-s − 0.499·6-s − 0.672·7-s + 1.08·8-s + 0.333·9-s − 0.557·10-s − 1.21·11-s − 0.145·12-s + 0.414·13-s + 0.581·14-s + 0.371·15-s − 0.685·16-s + 0.242·17-s − 0.288·18-s + 0.224·19-s − 0.161·20-s − 0.388·21-s + 1.05·22-s + 0.621·23-s + 0.625·24-s − 0.585·25-s − 0.358·26-s + 0.192·27-s + 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281351705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281351705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 19 | \( 1 - 0.978T + 19T^{2} \) |
| 23 | \( 1 - 2.98T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 0.00295T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 3.00T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 + 8.54T + 79T^{2} \) |
| 83 | \( 1 - 2.65T + 83T^{2} \) |
| 89 | \( 1 + 2.75T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−7.937332923984539296835133746441, −7.45915439268914148934800346569, −6.55879989367374053134467457552, −5.83888900115170910199236198374, −5.01083370338192457215840536127, −4.31847462458172305058995926896, −3.27206574692217126419609511761, −2.62184019656592052637643356941, −1.65310023809036769347158154304, −0.63596894918285434155400490435,
0.63596894918285434155400490435, 1.65310023809036769347158154304, 2.62184019656592052637643356941, 3.27206574692217126419609511761, 4.31847462458172305058995926896, 5.01083370338192457215840536127, 5.83888900115170910199236198374, 6.55879989367374053134467457552, 7.45915439268914148934800346569, 7.937332923984539296835133746441
