L(s) = 1 | − 1.66·2-s + 0.783·4-s − 3.76·5-s − 7-s + 2.02·8-s + 6.28·10-s + 3.16·11-s − 3.02·13-s + 1.66·14-s − 4.95·16-s − 1.05·17-s + 3.14·19-s − 2.95·20-s − 5.27·22-s + 7.12·23-s + 9.17·25-s + 5.04·26-s − 0.783·28-s − 2.67·29-s − 8.88·31-s + 4.20·32-s + 1.76·34-s + 3.76·35-s + 0.943·37-s − 5.24·38-s − 7.64·40-s − 9.90·41-s + ⋯ |
L(s) = 1 | − 1.17·2-s + 0.391·4-s − 1.68·5-s − 0.377·7-s + 0.717·8-s + 1.98·10-s + 0.953·11-s − 0.838·13-s + 0.445·14-s − 1.23·16-s − 0.256·17-s + 0.721·19-s − 0.659·20-s − 1.12·22-s + 1.48·23-s + 1.83·25-s + 0.988·26-s − 0.148·28-s − 0.496·29-s − 1.59·31-s + 0.743·32-s + 0.302·34-s + 0.636·35-s + 0.155·37-s − 0.851·38-s − 1.20·40-s − 1.54·41-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 | \( 1 + 1.66T + 2T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 + 8.88T + 31T^{2} \) |
| 37 | \( 1 - 0.943T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−7.41053954282002045335359451413, −7.19303162262208333705666010462, −6.60186732222091925743973492239, −5.16540887924653354364908211184, −4.69066491963206788947757412765, −3.69033529662713258187672351156, −3.31120538706073972481402844878, −1.93136039750661812946764356321, −0.847737433780786326464855758232, 0,
0.847737433780786326464855758232, 1.93136039750661812946764356321, 3.31120538706073972481402844878, 3.69033529662713258187672351156, 4.69066491963206788947757412765, 5.16540887924653354364908211184, 6.60186732222091925743973492239, 7.19303162262208333705666010462, 7.41053954282002045335359451413