| L(s) = 1 | + 2.69·2-s + 5.26·4-s − 1.47·5-s + 7-s + 8.80·8-s − 3.97·10-s + 3.56·11-s + 0.225·13-s + 2.69·14-s + 13.2·16-s − 1.78·17-s + 4.11·19-s − 7.76·20-s + 9.60·22-s + 2.13·23-s − 2.82·25-s + 0.609·26-s + 5.26·28-s + 3.51·29-s − 2.58·31-s + 18.0·32-s − 4.81·34-s − 1.47·35-s + 1.05·37-s + 11.0·38-s − 12.9·40-s − 1.46·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s − 0.659·5-s + 0.377·7-s + 3.11·8-s − 1.25·10-s + 1.07·11-s + 0.0626·13-s + 0.720·14-s + 3.30·16-s − 0.433·17-s + 0.943·19-s − 1.73·20-s + 2.04·22-s + 0.445·23-s − 0.565·25-s + 0.119·26-s + 0.995·28-s + 0.652·29-s − 0.464·31-s + 3.18·32-s − 0.826·34-s − 0.249·35-s + 0.173·37-s + 1.79·38-s − 2.05·40-s − 0.228·41-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
| \(L(1)\) |
\(\approx\) |
\(8.079917082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.079917082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 0.225T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 + 0.703T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 7.39T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.56T + 79T^{2} \) |
| 83 | \( 1 - 3.07T + 83T^{2} \) |
| 89 | \( 1 - 6.29T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.54387640027565099792087969014, −6.89675082731922894882000016873, −6.36009575769071518437780832389, −5.57191084149873352350246804150, −4.92482141648960567743557238942, −4.23287857237950844693371304850, −3.73083763558213004358788522902, −3.03569245503113186440694605647, −2.09134989509520909839454505086, −1.14007739183670673685822199274,
1.14007739183670673685822199274, 2.09134989509520909839454505086, 3.03569245503113186440694605647, 3.73083763558213004358788522902, 4.23287857237950844693371304850, 4.92482141648960567743557238942, 5.57191084149873352350246804150, 6.36009575769071518437780832389, 6.89675082731922894882000016873, 7.54387640027565099792087969014
