| L(s) = 1 | − 1.98·2-s + 1.92·4-s + 1.61·5-s + 7-s + 0.155·8-s − 3.19·10-s + 3.80·11-s + 3.75·13-s − 1.98·14-s − 4.15·16-s + 1.43·17-s − 3.83·19-s + 3.10·20-s − 7.53·22-s + 7.74·23-s − 2.39·25-s − 7.44·26-s + 1.92·28-s + 1.46·29-s + 7.16·31-s + 7.90·32-s − 2.84·34-s + 1.61·35-s + 1.16·37-s + 7.59·38-s + 0.250·40-s + 2.32·41-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 0.960·4-s + 0.722·5-s + 0.377·7-s + 0.0548·8-s − 1.01·10-s + 1.14·11-s + 1.04·13-s − 0.529·14-s − 1.03·16-s + 0.348·17-s − 0.879·19-s + 0.693·20-s − 1.60·22-s + 1.61·23-s − 0.478·25-s − 1.45·26-s + 0.363·28-s + 0.271·29-s + 1.28·31-s + 1.39·32-s − 0.487·34-s + 0.272·35-s + 0.192·37-s + 1.23·38-s + 0.0395·40-s + 0.363·41-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
| \(L(1)\) |
\(\approx\) |
\(1.516996342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516996342\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.98T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 - 7.16T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 - 5.68T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 0.528T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.031796398142624947545018611554, −7.29657764864002666305174720054, −6.43321279862654343878875541853, −6.21196094389482239749293226736, −5.04145757617178886533331888678, −4.32867614817509418481055991008, −3.36668335363746909135725553735, −2.25428613248775999429258415401, −1.45162707686168242195710382791, −0.858451393359907981109459486708,
0.858451393359907981109459486708, 1.45162707686168242195710382791, 2.25428613248775999429258415401, 3.36668335363746909135725553735, 4.32867614817509418481055991008, 5.04145757617178886533331888678, 6.21196094389482239749293226736, 6.43321279862654343878875541853, 7.29657764864002666305174720054, 8.031796398142624947545018611554
