L(s) = 1 | − 0.665·2-s − 1.55·4-s + 3.70·5-s + 7-s + 2.36·8-s − 2.46·10-s − 1.00·11-s + 3.77·13-s − 0.665·14-s + 1.53·16-s − 4.06·17-s − 8.44·19-s − 5.76·20-s + 0.667·22-s − 0.442·23-s + 8.72·25-s − 2.51·26-s − 1.55·28-s − 5.65·29-s + 8.26·31-s − 5.75·32-s + 2.70·34-s + 3.70·35-s − 10.2·37-s + 5.61·38-s + 8.76·40-s + 5.95·41-s + ⋯ |
L(s) = 1 | − 0.470·2-s − 0.778·4-s + 1.65·5-s + 0.377·7-s + 0.836·8-s − 0.779·10-s − 0.302·11-s + 1.04·13-s − 0.177·14-s + 0.384·16-s − 0.986·17-s − 1.93·19-s − 1.28·20-s + 0.142·22-s − 0.0922·23-s + 1.74·25-s − 0.492·26-s − 0.294·28-s − 1.04·29-s + 1.48·31-s − 1.01·32-s + 0.464·34-s + 0.626·35-s − 1.68·37-s + 0.910·38-s + 1.38·40-s + 0.930·41-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
\(L(1)\) |
\(\approx\) |
\(1.778058750\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.778058750\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.665T + 2T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 + 8.44T + 19T^{2} \) |
| 23 | \( 1 + 0.442T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 0.363T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 3.57T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 6.24T + 97T^{2} \) |
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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
−8.182582882599825908748844750400, −7.08576069632648970325685851405, −6.32643471144037313258455813191, −5.86004482572309278797490398110, −5.02959792963197206946937702840, −4.44340258380957855131926548599, −3.58451846100000807505784501176, −2.22095702839892115641190994232, −1.86609262865125613733798652900, −0.72200862126562241812089895384,
0.72200862126562241812089895384, 1.86609262865125613733798652900, 2.22095702839892115641190994232, 3.58451846100000807505784501176, 4.44340258380957855131926548599, 5.02959792963197206946937702840, 5.86004482572309278797490398110, 6.32643471144037313258455813191, 7.08576069632648970325685851405, 8.182582882599825908748844750400