| L(s) = 1 | − 1.55·2-s + 0.427·4-s − 3.44·5-s − 7-s + 2.44·8-s + 5.36·10-s + 2.27·11-s + 3.98·13-s + 1.55·14-s − 4.67·16-s + 0.0529·17-s + 1.72·19-s − 1.47·20-s − 3.55·22-s + 1.87·23-s + 6.87·25-s − 6.21·26-s − 0.427·28-s + 7.03·29-s + 9.09·31-s + 2.38·32-s − 0.0824·34-s + 3.44·35-s + 6.34·37-s − 2.68·38-s − 8.44·40-s + 4.26·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.213·4-s − 1.54·5-s − 0.377·7-s + 0.866·8-s + 1.69·10-s + 0.687·11-s + 1.10·13-s + 0.416·14-s − 1.16·16-s + 0.0128·17-s + 0.394·19-s − 0.329·20-s − 0.756·22-s + 0.391·23-s + 1.37·25-s − 1.21·26-s − 0.0808·28-s + 1.30·29-s + 1.63·31-s + 0.420·32-s − 0.0141·34-s + 0.582·35-s + 1.04·37-s − 0.434·38-s − 1.33·40-s + 0.665·41-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
| \(L(1)\) |
\(\approx\) |
\(0.8675482179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8675482179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.55T + 2T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 3.98T + 13T^{2} \) |
| 17 | \( 1 - 0.0529T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 + 0.634T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 + 1.35T + 97T^{2} \) |
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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
−8.088243146787283024774406949482, −7.36185143254431326366671635619, −6.70124426447531956128012136856, −6.04086389954482958590982619406, −4.73901748068835278712291082625, −4.25663305073248546805488449973, −3.56522303274377667993369426967, −2.69548310608162203362206354314, −1.17138426198313992540187830601, −0.68183171476594379767753295628,
0.68183171476594379767753295628, 1.17138426198313992540187830601, 2.69548310608162203362206354314, 3.56522303274377667993369426967, 4.25663305073248546805488449973, 4.73901748068835278712291082625, 6.04086389954482958590982619406, 6.70124426447531956128012136856, 7.36185143254431326366671635619, 8.088243146787283024774406949482
