| L(s) = 1 | + 2.26·2-s + 3.13·4-s + 1.21·5-s + 7-s + 2.58·8-s + 2.76·10-s + 0.403·11-s + 6.29·13-s + 2.26·14-s − 0.421·16-s + 3.86·17-s + 4.73·19-s + 3.82·20-s + 0.914·22-s − 5.88·23-s − 3.51·25-s + 14.2·26-s + 3.13·28-s − 3.62·29-s − 1.66·31-s − 6.12·32-s + 8.75·34-s + 1.21·35-s + 9.01·37-s + 10.7·38-s + 3.14·40-s + 7.34·41-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.56·4-s + 0.544·5-s + 0.377·7-s + 0.913·8-s + 0.873·10-s + 0.121·11-s + 1.74·13-s + 0.605·14-s − 0.105·16-s + 0.936·17-s + 1.08·19-s + 0.855·20-s + 0.195·22-s − 1.22·23-s − 0.703·25-s + 2.79·26-s + 0.593·28-s − 0.673·29-s − 0.298·31-s − 1.08·32-s + 1.50·34-s + 0.205·35-s + 1.48·37-s + 1.74·38-s + 0.497·40-s + 1.14·41-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
| \(L(1)\) |
\(\approx\) |
\(7.215100795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.215100795\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 11 | \( 1 - 0.403T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 9.01T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 - 9.42T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 - 7.97T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 9.51T + 97T^{2} \) |
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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
−7.82975922629619175812267309767, −6.79763839102678368481010397881, −6.08984090771001795003435030075, −5.69175673021328884508218097370, −5.20566250403994860285526519211, −4.04502226329579447121710847430, −3.82751832932379501548554578172, −2.92266365166210227880191580428, −2.00575822504305245922268774221, −1.13680203980561931572529754211,
1.13680203980561931572529754211, 2.00575822504305245922268774221, 2.92266365166210227880191580428, 3.82751832932379501548554578172, 4.04502226329579447121710847430, 5.20566250403994860285526519211, 5.69175673021328884508218097370, 6.08984090771001795003435030075, 6.79763839102678368481010397881, 7.82975922629619175812267309767
