| L(s) = 1 | − 2.68·2-s + 5.21·4-s − 3.72·5-s + 7-s − 8.62·8-s + 10.0·10-s + 4.70·11-s − 1.94·13-s − 2.68·14-s + 12.7·16-s − 6.80·17-s − 1.46·19-s − 19.4·20-s − 12.6·22-s − 1.68·23-s + 8.87·25-s + 5.21·26-s + 5.21·28-s + 5.63·29-s + 8.82·31-s − 16.9·32-s + 18.2·34-s − 3.72·35-s − 10.5·37-s + 3.93·38-s + 32.1·40-s − 10.3·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.60·4-s − 1.66·5-s + 0.377·7-s − 3.04·8-s + 3.16·10-s + 1.41·11-s − 0.539·13-s − 0.717·14-s + 3.18·16-s − 1.64·17-s − 0.335·19-s − 4.34·20-s − 2.69·22-s − 0.350·23-s + 1.77·25-s + 1.02·26-s + 0.985·28-s + 1.04·29-s + 1.58·31-s − 2.99·32-s + 3.13·34-s − 0.629·35-s − 1.73·37-s + 0.637·38-s + 5.08·40-s − 1.61·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3486306617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3486306617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 + 3.72T + 5T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 7.92T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 0.628T + 83T^{2} \) |
| 89 | \( 1 + 5.11T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−8.132340245407307760037308140915, −7.23367261264799822743101764298, −6.76309646696920073902912905407, −6.44477752892345421054693980737, −4.91854413407700230503147317275, −4.16043021133144764033298394571, −3.34154669430921291363980421646, −2.34712466608388554722623258634, −1.42744444806898979366044903516, −0.40863280128540623714363301707,
0.40863280128540623714363301707, 1.42744444806898979366044903516, 2.34712466608388554722623258634, 3.34154669430921291363980421646, 4.16043021133144764033298394571, 4.91854413407700230503147317275, 6.44477752892345421054693980737, 6.76309646696920073902912905407, 7.23367261264799822743101764298, 8.132340245407307760037308140915
