| L(s) = 1 | − 2.31·2-s − 0.975·3-s + 3.35·4-s − 3.23·5-s + 2.25·6-s + 5.09·7-s − 3.14·8-s − 2.04·9-s + 7.47·10-s + 3.94·11-s − 3.27·12-s + 1.47·13-s − 11.7·14-s + 3.15·15-s + 0.565·16-s + 17-s + 4.74·18-s + 0.961·19-s − 10.8·20-s − 4.96·21-s − 9.14·22-s + 0.0822·23-s + 3.06·24-s + 5.43·25-s − 3.41·26-s + 4.92·27-s + 17.1·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.562·3-s + 1.67·4-s − 1.44·5-s + 0.921·6-s + 1.92·7-s − 1.11·8-s − 0.683·9-s + 2.36·10-s + 1.19·11-s − 0.945·12-s + 0.409·13-s − 3.15·14-s + 0.813·15-s + 0.141·16-s + 0.242·17-s + 1.11·18-s + 0.220·19-s − 2.42·20-s − 1.08·21-s − 1.94·22-s + 0.0171·23-s + 0.626·24-s + 1.08·25-s − 0.670·26-s + 0.947·27-s + 3.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 0.975T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 19 | \( 1 - 0.961T + 19T^{2} \) |
| 23 | \( 1 - 0.0822T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 4.26T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 7.16T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 9.02T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.039305517944810514676414422293, −7.25259583470462881608630561564, −6.73263483497439378092709104162, −5.68722756806493738642400043360, −4.78686059472751328954382179600, −4.13853354394603018319134516392, −3.08504559671358592538840449240, −1.70010184677534548117803447308, −1.11586482626326427059863755255, 0,
1.11586482626326427059863755255, 1.70010184677534548117803447308, 3.08504559671358592538840449240, 4.13853354394603018319134516392, 4.78686059472751328954382179600, 5.68722756806493738642400043360, 6.73263483497439378092709104162, 7.25259583470462881608630561564, 8.039305517944810514676414422293
