| L(s) = 1 | + 1.77·2-s + 1.13·4-s − 1.38·5-s + 7-s − 1.53·8-s − 2.45·10-s − 3.51·11-s + 4.71·13-s + 1.77·14-s − 4.98·16-s + 4.68·17-s − 3.74·19-s − 1.57·20-s − 6.22·22-s + 2.00·23-s − 3.08·25-s + 8.34·26-s + 1.13·28-s + 4.37·29-s − 0.875·31-s − 5.76·32-s + 8.29·34-s − 1.38·35-s − 9.02·37-s − 6.63·38-s + 2.11·40-s − 1.57·41-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.567·4-s − 0.618·5-s + 0.377·7-s − 0.541·8-s − 0.774·10-s − 1.05·11-s + 1.30·13-s + 0.473·14-s − 1.24·16-s + 1.13·17-s − 0.859·19-s − 0.351·20-s − 1.32·22-s + 0.417·23-s − 0.617·25-s + 1.63·26-s + 0.214·28-s + 0.811·29-s − 0.157·31-s − 1.01·32-s + 1.42·34-s − 0.233·35-s − 1.48·37-s − 1.07·38-s + 0.334·40-s − 0.245·41-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
| \(L(1)\) |
\(\approx\) |
\(3.063473281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.063473281\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.77T + 2T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 0.875T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 0.801T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 8.91T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.05T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−7.84027751423313253178876369611, −6.98060499421262676447568539040, −6.28317136802977571892806226409, −5.39481739972564486148451189829, −5.22229821597916534479593922985, −4.10636640003628164879611873461, −3.76656273351828497569216230036, −2.97511540210777780391486860823, −2.07197502041433505216774095577, −0.70513061146608351961765347578,
0.70513061146608351961765347578, 2.07197502041433505216774095577, 2.97511540210777780391486860823, 3.76656273351828497569216230036, 4.10636640003628164879611873461, 5.22229821597916534479593922985, 5.39481739972564486148451189829, 6.28317136802977571892806226409, 6.98060499421262676447568539040, 7.84027751423313253178876369611
