| L(s) = 1 | − 1.07·2-s − 0.842·4-s + 1.06·5-s + 7-s + 3.05·8-s − 1.14·10-s + 1.63·11-s + 2.87·13-s − 1.07·14-s − 1.60·16-s + 0.537·17-s + 3.98·19-s − 0.900·20-s − 1.75·22-s + 7.38·23-s − 3.85·25-s − 3.09·26-s − 0.842·28-s + 4.43·29-s + 7.15·31-s − 4.39·32-s − 0.578·34-s + 1.06·35-s + 2.70·37-s − 4.28·38-s + 3.26·40-s + 6.22·41-s + ⋯ |
| L(s) = 1 | − 0.760·2-s − 0.421·4-s + 0.478·5-s + 0.377·7-s + 1.08·8-s − 0.363·10-s + 0.493·11-s + 0.797·13-s − 0.287·14-s − 0.401·16-s + 0.130·17-s + 0.914·19-s − 0.201·20-s − 0.375·22-s + 1.54·23-s − 0.771·25-s − 0.606·26-s − 0.159·28-s + 0.822·29-s + 1.28·31-s − 0.776·32-s − 0.0991·34-s + 0.180·35-s + 0.444·37-s − 0.695·38-s + 0.516·40-s + 0.971·41-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
| \(L(1)\) |
\(\approx\) |
\(1.770892399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770892399\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.07T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 0.537T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 9.71T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 3.30T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 9.32T + 97T^{2} \) |
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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
−8.133655743650794881447049092017, −7.21200554149584731049623807737, −6.61709120441871676681829733238, −5.71532032710311491396092586379, −5.06057271072860003142325262210, −4.34166228417518931676634855199, −3.53347236424073954213889755498, −2.52837332113645084082666652365, −1.35844985174913482655287369293, −0.883385201612521380999122425161,
0.883385201612521380999122425161, 1.35844985174913482655287369293, 2.52837332113645084082666652365, 3.53347236424073954213889755498, 4.34166228417518931676634855199, 5.06057271072860003142325262210, 5.71532032710311491396092586379, 6.61709120441871676681829733238, 7.21200554149584731049623807737, 8.133655743650794881447049092017
