| L(s) = 1 | + 0.782·2-s − 1.38·4-s + 2.85·5-s + 7-s − 2.65·8-s + 2.23·10-s + 5.95·11-s − 2.24·13-s + 0.782·14-s + 0.702·16-s − 0.355·17-s + 4.58·19-s − 3.96·20-s + 4.65·22-s − 1.77·23-s + 3.15·25-s − 1.75·26-s − 1.38·28-s + 9.18·29-s − 2.61·31-s + 5.85·32-s − 0.277·34-s + 2.85·35-s + 0.446·37-s + 3.58·38-s − 7.57·40-s + 5.55·41-s + ⋯ |
| L(s) = 1 | + 0.553·2-s − 0.693·4-s + 1.27·5-s + 0.377·7-s − 0.937·8-s + 0.706·10-s + 1.79·11-s − 0.622·13-s + 0.209·14-s + 0.175·16-s − 0.0861·17-s + 1.05·19-s − 0.886·20-s + 0.992·22-s − 0.369·23-s + 0.631·25-s − 0.344·26-s − 0.262·28-s + 1.70·29-s − 0.470·31-s + 1.03·32-s − 0.0476·34-s + 0.482·35-s + 0.0733·37-s + 0.581·38-s − 1.19·40-s + 0.867·41-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
| \(L(1)\) |
\(\approx\) |
\(3.470509064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.470509064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.782T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 0.355T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 0.446T + 37T^{2} \) |
| 41 | \( 1 - 5.55T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−7.85618614871023753635834776479, −6.90992240527028914760215667814, −6.24787115502401918449277532790, −5.76206398085447728489872815724, −4.96186806640141548118777683673, −4.44850851711856005433288831091, −3.60496948952144142748088975025, −2.76997627029649222235061726926, −1.74300312463644886952661318779, −0.914498046050812368305818817012,
0.914498046050812368305818817012, 1.74300312463644886952661318779, 2.76997627029649222235061726926, 3.60496948952144142748088975025, 4.44850851711856005433288831091, 4.96186806640141548118777683673, 5.76206398085447728489872815724, 6.24787115502401918449277532790, 6.90992240527028914760215667814, 7.85618614871023753635834776479
