| L(s) = 1 | − 3-s + 3.09·5-s + 2.06·7-s + 9-s + 4.75·11-s + 0.499·13-s − 3.09·15-s + 6.36·17-s − 6.44·19-s − 2.06·21-s − 0.293·23-s + 4.60·25-s − 27-s + 7.05·29-s + 1.77·31-s − 4.75·33-s + 6.41·35-s − 3.97·37-s − 0.499·39-s − 2.87·41-s + 11.0·43-s + 3.09·45-s + 1.77·47-s − 2.71·49-s − 6.36·51-s + 1.45·53-s + 14.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.38·5-s + 0.782·7-s + 0.333·9-s + 1.43·11-s + 0.138·13-s − 0.800·15-s + 1.54·17-s − 1.47·19-s − 0.451·21-s − 0.0611·23-s + 0.920·25-s − 0.192·27-s + 1.31·29-s + 0.318·31-s − 0.828·33-s + 1.08·35-s − 0.652·37-s − 0.0799·39-s − 0.448·41-s + 1.69·43-s + 0.461·45-s + 0.258·47-s − 0.388·49-s − 0.890·51-s + 0.199·53-s + 1.98·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110533790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110533790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 - 0.499T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 0.293T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 + 9.16T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−7.80152814156942453430547231888, −6.95571926560652892445846532385, −6.17976810837472484515502518374, −5.99020238454921035998459541117, −5.08769821264120492660069273817, −4.46052877096405260498294974859, −3.61112299381320912148199102441, −2.44331315239776095956037253440, −1.59852927481738803540598065497, −1.01538740292530783683791403304,
1.01538740292530783683791403304, 1.59852927481738803540598065497, 2.44331315239776095956037253440, 3.61112299381320912148199102441, 4.46052877096405260498294974859, 5.08769821264120492660069273817, 5.99020238454921035998459541117, 6.17976810837472484515502518374, 6.95571926560652892445846532385, 7.80152814156942453430547231888
