| L(s) = 1 | − 0.301·2-s − 1.90·4-s − 3.64·5-s + 7-s + 1.17·8-s + 1.09·10-s − 2.79·11-s + 5.84·13-s − 0.301·14-s + 3.46·16-s + 5.05·17-s + 5.17·19-s + 6.95·20-s + 0.842·22-s + 7.69·23-s + 8.28·25-s − 1.76·26-s − 1.90·28-s − 5.38·29-s + 8.19·31-s − 3.40·32-s − 1.52·34-s − 3.64·35-s − 10.6·37-s − 1.56·38-s − 4.29·40-s + 7.27·41-s + ⋯ |
| L(s) = 1 | − 0.213·2-s − 0.954·4-s − 1.62·5-s + 0.377·7-s + 0.416·8-s + 0.347·10-s − 0.843·11-s + 1.62·13-s − 0.0805·14-s + 0.865·16-s + 1.22·17-s + 1.18·19-s + 1.55·20-s + 0.179·22-s + 1.60·23-s + 1.65·25-s − 0.345·26-s − 0.360·28-s − 1.00·29-s + 1.47·31-s − 0.601·32-s − 0.261·34-s − 0.616·35-s − 1.75·37-s − 0.253·38-s − 0.679·40-s + 1.13·41-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
| \(L(1)\) |
\(\approx\) |
\(1.170451788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170451788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.301T + 2T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.730T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 0.0983T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−7.960260376107733124973110085315, −7.51628796976516290174120033654, −6.59631383039694291244040488155, −5.40686450085509020694833541989, −5.08806485759481307571181360745, −4.22855062021688864414357576482, −3.43574903231299498048741135024, −3.17453062320331250614301635625, −1.30110401588250276831600665041, −0.65549122232490501181521727424,
0.65549122232490501181521727424, 1.30110401588250276831600665041, 3.17453062320331250614301635625, 3.43574903231299498048741135024, 4.22855062021688864414357576482, 5.08806485759481307571181360745, 5.40686450085509020694833541989, 6.59631383039694291244040488155, 7.51628796976516290174120033654, 7.960260376107733124973110085315
