L(s) = 1 | − 0.903·2-s − 1.18·4-s + 2.97·5-s − 7-s + 2.87·8-s − 2.69·10-s + 0.876·11-s − 1.42·13-s + 0.903·14-s − 0.235·16-s − 0.579·17-s + 7.77·19-s − 3.52·20-s − 0.792·22-s − 3.51·23-s + 3.87·25-s + 1.28·26-s + 1.18·28-s + 1.75·29-s − 5.47·31-s − 5.54·32-s + 0.524·34-s − 2.97·35-s − 8.19·37-s − 7.02·38-s + 8.57·40-s − 5.68·41-s + ⋯ |
L(s) = 1 | − 0.639·2-s − 0.591·4-s + 1.33·5-s − 0.377·7-s + 1.01·8-s − 0.851·10-s + 0.264·11-s − 0.395·13-s + 0.241·14-s − 0.0588·16-s − 0.140·17-s + 1.78·19-s − 0.787·20-s − 0.168·22-s − 0.733·23-s + 0.774·25-s + 0.252·26-s + 0.223·28-s + 0.324·29-s − 0.983·31-s − 0.979·32-s + 0.0898·34-s − 0.503·35-s − 1.34·37-s − 1.14·38-s + 1.35·40-s − 0.888·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.903T + 2T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 11 | \( 1 - 0.876T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 + 0.579T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 23 | \( 1 + 3.51T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 + 8.19T + 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 7.37T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 + 8.42T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 0.702T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.62123241163707561200195678104, −6.82276747019592155892449828511, −6.14186913382628091967508628757, −5.28146884315531824062247665467, −4.97259615556462860123787734358, −3.82290328716999077081811872980, −3.05807440391141445603321385279, −1.92347726644032872591887918533, −1.30113546404462336011615999019, 0,
1.30113546404462336011615999019, 1.92347726644032872591887918533, 3.05807440391141445603321385279, 3.82290328716999077081811872980, 4.97259615556462860123787734358, 5.28146884315531824062247665467, 6.14186913382628091967508628757, 6.82276747019592155892449828511, 7.62123241163707561200195678104