| L(s) = 1 | + 2-s + 4-s − 2.10·5-s + 2.35·7-s + 8-s − 2.10·10-s + 1.36·11-s + 6.92·13-s + 2.35·14-s + 16-s + 0.171·17-s − 6.80·19-s − 2.10·20-s + 1.36·22-s + 1.85·23-s − 0.561·25-s + 6.92·26-s + 2.35·28-s + 1.03·29-s + 7.57·31-s + 32-s + 0.171·34-s − 4.96·35-s + 0.829·37-s − 6.80·38-s − 2.10·40-s + 4.45·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.942·5-s + 0.890·7-s + 0.353·8-s − 0.666·10-s + 0.410·11-s + 1.92·13-s + 0.629·14-s + 0.250·16-s + 0.0415·17-s − 1.56·19-s − 0.471·20-s + 0.290·22-s + 0.387·23-s − 0.112·25-s + 1.35·26-s + 0.445·28-s + 0.192·29-s + 1.36·31-s + 0.176·32-s + 0.0293·34-s − 0.838·35-s + 0.136·37-s − 1.10·38-s − 0.333·40-s + 0.695·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.193653436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.193653436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 2.10T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 - 0.829T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 + 9.80T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.219355246905203411334493489529, −7.950365572616889222619098533203, −6.71268455939428794184552312098, −6.34814647044547067004825525438, −5.38713324470481186763302216370, −4.40144073265270413069969750013, −4.05011413349115119271882829557, −3.22601611311323799164892757134, −2.01166449519229385263137595724, −0.976809463328849346527502799614,
0.976809463328849346527502799614, 2.01166449519229385263137595724, 3.22601611311323799164892757134, 4.05011413349115119271882829557, 4.40144073265270413069969750013, 5.38713324470481186763302216370, 6.34814647044547067004825525438, 6.71268455939428794184552312098, 7.950365572616889222619098533203, 8.219355246905203411334493489529
