| L(s) = 1 | + 0.249·2-s − 1.93·4-s − 0.989·5-s + 7-s − 0.981·8-s − 0.246·10-s + 2.01·11-s + 4.18·13-s + 0.249·14-s + 3.63·16-s + 7.51·17-s + 7.64·19-s + 1.91·20-s + 0.503·22-s + 2.82·23-s − 4.02·25-s + 1.04·26-s − 1.93·28-s − 5.08·29-s + 1.14·31-s + 2.86·32-s + 1.87·34-s − 0.989·35-s + 10.6·37-s + 1.90·38-s + 0.971·40-s − 8.51·41-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.968·4-s − 0.442·5-s + 0.377·7-s − 0.346·8-s − 0.0779·10-s + 0.608·11-s + 1.15·13-s + 0.0665·14-s + 0.907·16-s + 1.82·17-s + 1.75·19-s + 0.428·20-s + 0.107·22-s + 0.589·23-s − 0.804·25-s + 0.204·26-s − 0.366·28-s − 0.944·29-s + 0.205·31-s + 0.506·32-s + 0.320·34-s − 0.167·35-s + 1.74·37-s + 0.309·38-s + 0.153·40-s − 1.33·41-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
| \(L(1)\) |
\(\approx\) |
\(2.184952272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184952272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.249T + 2T^{2} \) |
| 5 | \( 1 + 0.989T + 5T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 - 0.347T + 47T^{2} \) |
| 53 | \( 1 - 4.33T + 53T^{2} \) |
| 59 | \( 1 - 9.40T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 4.93T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.889592633890453672732673827384, −7.37037581375330684180995042443, −6.31211944004051909252853694082, −5.49507595730850713900346944675, −5.21515407029071671168071637393, −4.09778166740747395470103818692, −3.65451532446080884738010189165, −3.04372378686940196456665110576, −1.41449337677926837026385971292, −0.834082657759181678977829722454,
0.834082657759181678977829722454, 1.41449337677926837026385971292, 3.04372378686940196456665110576, 3.65451532446080884738010189165, 4.09778166740747395470103818692, 5.21515407029071671168071637393, 5.49507595730850713900346944675, 6.31211944004051909252853694082, 7.37037581375330684180995042443, 7.889592633890453672732673827384
