| L(s) = 1 | − 2.72·2-s + 5.42·4-s + 0.150·5-s − 4.90·7-s − 9.34·8-s − 0.409·10-s + 0.576·11-s + 2.66·13-s + 13.3·14-s + 14.6·16-s + 5.36·17-s − 19-s + 0.816·20-s − 1.57·22-s + 6.79·23-s − 4.97·25-s − 7.27·26-s − 26.6·28-s − 0.917·29-s + 7.50·31-s − 21.1·32-s − 14.6·34-s − 0.738·35-s + 5.23·37-s + 2.72·38-s − 1.40·40-s − 6.67·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.71·4-s + 0.0672·5-s − 1.85·7-s − 3.30·8-s − 0.129·10-s + 0.173·11-s + 0.739·13-s + 3.57·14-s + 3.65·16-s + 1.30·17-s − 0.229·19-s + 0.182·20-s − 0.335·22-s + 1.41·23-s − 0.995·25-s − 1.42·26-s − 5.03·28-s − 0.170·29-s + 1.34·31-s − 3.73·32-s − 2.50·34-s − 0.124·35-s + 0.860·37-s + 0.442·38-s − 0.222·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6411765180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6411765180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 0.150T + 5T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 - 0.576T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 23 | \( 1 - 6.79T + 23T^{2} \) |
| 29 | \( 1 + 0.917T + 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 + 5.85T + 59T^{2} \) |
| 61 | \( 1 + 7.05T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 + 8.02T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−7.916232382745738040609823211154, −7.27474134614383740780136133298, −6.60261904679961938393770771668, −6.17909460900789411224863128953, −5.52790266090352156334015974501, −3.87636012516358752488363168393, −3.11239592767849046312771642810, −2.59663890351193055519586384849, −1.32689707658538292098089010346, −0.57993301404207758911455953099,
0.57993301404207758911455953099, 1.32689707658538292098089010346, 2.59663890351193055519586384849, 3.11239592767849046312771642810, 3.87636012516358752488363168393, 5.52790266090352156334015974501, 6.17909460900789411224863128953, 6.60261904679961938393770771668, 7.27474134614383740780136133298, 7.916232382745738040609823211154
