| L(s) = 1 | + 0.102·2-s − 1.98·4-s − 0.280·5-s − 7-s − 0.407·8-s − 0.0286·10-s − 0.974·11-s − 1.41·13-s − 0.102·14-s + 3.93·16-s − 6.28·17-s + 1.00·19-s + 0.558·20-s − 0.0996·22-s − 1.39·23-s − 4.92·25-s − 0.144·26-s + 1.98·28-s + 2.24·29-s − 9.08·31-s + 1.21·32-s − 0.642·34-s + 0.280·35-s + 8.73·37-s + 0.102·38-s + 0.114·40-s + 3.37·41-s + ⋯ |
| L(s) = 1 | + 0.0722·2-s − 0.994·4-s − 0.125·5-s − 0.377·7-s − 0.144·8-s − 0.00907·10-s − 0.293·11-s − 0.392·13-s − 0.0273·14-s + 0.984·16-s − 1.52·17-s + 0.231·19-s + 0.124·20-s − 0.0212·22-s − 0.290·23-s − 0.984·25-s − 0.0283·26-s + 0.375·28-s + 0.416·29-s − 1.63·31-s + 0.215·32-s − 0.110·34-s + 0.0474·35-s + 1.43·37-s + 0.0167·38-s + 0.0180·40-s + 0.527·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6077202019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6077202019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.102T + 2T^{2} \) |
| 5 | \( 1 + 0.280T + 5T^{2} \) |
| 11 | \( 1 + 0.974T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 1.77T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−7.74060530952583982068803944458, −7.35435029411366189633355641438, −6.26640360434330709384016185241, −5.80944405883292889890928025336, −4.85419043628292286274312097119, −4.37467821174608381006374801437, −3.63472222083740372329324986362, −2.76801029149879019838906254205, −1.77638419587622674279925468556, −0.37308867443356982644833260876,
0.37308867443356982644833260876, 1.77638419587622674279925468556, 2.76801029149879019838906254205, 3.63472222083740372329324986362, 4.37467821174608381006374801437, 4.85419043628292286274312097119, 5.80944405883292889890928025336, 6.26640360434330709384016185241, 7.35435029411366189633355641438, 7.74060530952583982068803944458
