| L(s) = 1 | + 1.24·2-s − 0.443·4-s − 1.34·5-s − 7-s − 3.04·8-s − 1.67·10-s + 5.34·11-s − 4.35·13-s − 1.24·14-s − 2.91·16-s + 3.15·17-s + 2.16·19-s + 0.595·20-s + 6.67·22-s − 6.10·23-s − 3.20·25-s − 5.42·26-s + 0.443·28-s + 1.86·29-s + 0.389·31-s + 2.46·32-s + 3.93·34-s + 1.34·35-s + 1.82·37-s + 2.70·38-s + 4.08·40-s − 4.04·41-s + ⋯ |
| L(s) = 1 | + 0.882·2-s − 0.221·4-s − 0.599·5-s − 0.377·7-s − 1.07·8-s − 0.529·10-s + 1.61·11-s − 1.20·13-s − 0.333·14-s − 0.728·16-s + 0.765·17-s + 0.496·19-s + 0.133·20-s + 1.42·22-s − 1.27·23-s − 0.640·25-s − 1.06·26-s + 0.0839·28-s + 0.347·29-s + 0.0700·31-s + 0.435·32-s + 0.675·34-s + 0.226·35-s + 0.299·37-s + 0.438·38-s + 0.646·40-s − 0.631·41-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
| \(L(1)\) |
\(\approx\) |
\(1.773698782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773698782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 - 0.389T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 - 0.687T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 2.01T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 + 1.44T + 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−7.85254375209219847350792469038, −6.92294695954109595648833972705, −6.42622685549850634796787641904, −5.62746787482783146053315209326, −4.95782650818962756099568285353, −4.16056738871759029683497686196, −3.70035547286332163578478302510, −3.05417963215981710674154225664, −1.90850001908175136073209009727, −0.56731512875469240854585715370,
0.56731512875469240854585715370, 1.90850001908175136073209009727, 3.05417963215981710674154225664, 3.70035547286332163578478302510, 4.16056738871759029683497686196, 4.95782650818962756099568285353, 5.62746787482783146053315209326, 6.42622685549850634796787641904, 6.92294695954109595648833972705, 7.85254375209219847350792469038
