| L(s) = 1 | − 2-s − 2.00·3-s + 4-s + 3.10·5-s + 2.00·6-s + 0.986·7-s − 8-s + 1.01·9-s − 3.10·10-s + 3.67·11-s − 2.00·12-s + 3.31·13-s − 0.986·14-s − 6.22·15-s + 16-s + 0.586·17-s − 1.01·18-s + 4.79·19-s + 3.10·20-s − 1.97·21-s − 3.67·22-s − 5.09·23-s + 2.00·24-s + 4.67·25-s − 3.31·26-s + 3.98·27-s + 0.986·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 0.5·4-s + 1.39·5-s + 0.817·6-s + 0.372·7-s − 0.353·8-s + 0.337·9-s − 0.983·10-s + 1.10·11-s − 0.578·12-s + 0.920·13-s − 0.263·14-s − 1.60·15-s + 0.250·16-s + 0.142·17-s − 0.238·18-s + 1.10·19-s + 0.695·20-s − 0.431·21-s − 0.783·22-s − 1.06·23-s + 0.408·24-s + 0.934·25-s − 0.650·26-s + 0.766·27-s + 0.186·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.703506133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703506133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 - 0.986T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 0.586T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 0.441T + 41T^{2} \) |
| 43 | \( 1 - 13.0T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 + 5.28T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−7.901416001319946929483903208625, −6.89379569177112711448934629067, −6.26751244002112067247324029604, −6.01020367966608051709558701304, −5.31569327712739512788429920010, −4.50971691061848996742632715361, −3.42274525859708631991818012755, −2.35633641029746014955292916164, −1.39747110669178010896596298836, −0.879539462459782590241787899313,
0.879539462459782590241787899313, 1.39747110669178010896596298836, 2.35633641029746014955292916164, 3.42274525859708631991818012755, 4.50971691061848996742632715361, 5.31569327712739512788429920010, 6.01020367966608051709558701304, 6.26751244002112067247324029604, 6.89379569177112711448934629067, 7.901416001319946929483903208625
