| L(s) = 1 | + 0.407·2-s − 1.83·4-s + 1.08·5-s + 7-s − 1.56·8-s + 0.444·10-s − 2.30·11-s − 1.28·13-s + 0.407·14-s + 3.02·16-s − 3.85·17-s − 5.48·19-s − 1.99·20-s − 0.938·22-s − 5.15·23-s − 3.81·25-s − 0.523·26-s − 1.83·28-s + 3.35·29-s − 0.261·31-s + 4.36·32-s − 1.57·34-s + 1.08·35-s + 4.76·37-s − 2.23·38-s − 1.70·40-s − 4.53·41-s + ⋯ |
| L(s) = 1 | + 0.288·2-s − 0.916·4-s + 0.487·5-s + 0.377·7-s − 0.552·8-s + 0.140·10-s − 0.693·11-s − 0.355·13-s + 0.109·14-s + 0.757·16-s − 0.936·17-s − 1.25·19-s − 0.446·20-s − 0.200·22-s − 1.07·23-s − 0.762·25-s − 0.102·26-s − 0.346·28-s + 0.622·29-s − 0.0470·31-s + 0.771·32-s − 0.270·34-s + 0.184·35-s + 0.783·37-s − 0.362·38-s − 0.269·40-s − 0.708·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253315246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253315246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.407T + 2T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 0.261T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 0.491T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 1.96T + 59T^{2} \) |
| 61 | \( 1 - 0.0291T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 - 9.68T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−8.188880660107996932404996862384, −7.10475256114005153255523492033, −6.25805521210753868924064134412, −5.70363634934105569543793381437, −4.95973888949513114572995586438, −4.35876713787497010301074816839, −3.74889404333134458962982874095, −2.54513614678883617539028117699, −1.97580459245144797338721717463, −0.51215162268467314258008314380,
0.51215162268467314258008314380, 1.97580459245144797338721717463, 2.54513614678883617539028117699, 3.74889404333134458962982874095, 4.35876713787497010301074816839, 4.95973888949513114572995586438, 5.70363634934105569543793381437, 6.25805521210753868924064134412, 7.10475256114005153255523492033, 8.188880660107996932404996862384
