| L(s) = 1 | + 2.51·2-s + 4.31·4-s − 4.05·5-s + 1.37·7-s + 5.82·8-s − 10.1·10-s + 0.0273·11-s − 0.627·13-s + 3.46·14-s + 6.00·16-s − 4.30·17-s − 5.00·19-s − 17.5·20-s + 0.0686·22-s + 11.4·25-s − 1.57·26-s + 5.94·28-s − 3.69·29-s − 0.482·31-s + 3.43·32-s − 10.8·34-s − 5.58·35-s + 2.20·37-s − 12.5·38-s − 23.6·40-s − 10.0·41-s + 3.28·43-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.15·4-s − 1.81·5-s + 0.520·7-s + 2.05·8-s − 3.22·10-s + 0.00823·11-s − 0.173·13-s + 0.924·14-s + 1.50·16-s − 1.04·17-s − 1.14·19-s − 3.91·20-s + 0.0146·22-s + 2.28·25-s − 0.309·26-s + 1.12·28-s − 0.686·29-s − 0.0866·31-s + 0.607·32-s − 1.85·34-s − 0.943·35-s + 0.362·37-s − 2.03·38-s − 3.73·40-s − 1.56·41-s + 0.500·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 4.05T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 0.0273T + 11T^{2} \) |
| 13 | \( 1 + 0.627T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 + 0.482T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 1.83T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 + 4.62T + 61T^{2} \) |
| 67 | \( 1 + 4.42T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 - 5.97T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2.16T + 97T^{2} \) |
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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
−7.83780726767064636703930181195, −6.84564943766112887328952193217, −6.61793576134925831295634803091, −5.42119517708182603264191189878, −4.70370491055110911500031204266, −4.23213875037214924324199902460, −3.65505455887020547546744618057, −2.82109756727689480773476315781, −1.81009402721287314188056961604, 0,
1.81009402721287314188056961604, 2.82109756727689480773476315781, 3.65505455887020547546744618057, 4.23213875037214924324199902460, 4.70370491055110911500031204266, 5.42119517708182603264191189878, 6.61793576134925831295634803091, 6.84564943766112887328952193217, 7.83780726767064636703930181195
