| L(s) = 1 | − 2.16·2-s − 0.113·3-s + 2.67·4-s + 5-s + 0.245·6-s − 2.73·7-s − 1.45·8-s − 2.98·9-s − 2.16·10-s + 1.80·11-s − 0.303·12-s + 6.30·13-s + 5.90·14-s − 0.113·15-s − 2.20·16-s − 2.83·17-s + 6.45·18-s + 0.0940·19-s + 2.67·20-s + 0.310·21-s − 3.90·22-s − 0.210·23-s + 0.165·24-s + 25-s − 13.6·26-s + 0.679·27-s − 7.30·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.0655·3-s + 1.33·4-s + 0.447·5-s + 0.100·6-s − 1.03·7-s − 0.514·8-s − 0.995·9-s − 0.683·10-s + 0.544·11-s − 0.0875·12-s + 1.74·13-s + 1.57·14-s − 0.0293·15-s − 0.550·16-s − 0.688·17-s + 1.52·18-s + 0.0215·19-s + 0.597·20-s + 0.0677·21-s − 0.831·22-s − 0.0438·23-s + 0.0337·24-s + 0.200·25-s − 2.67·26-s + 0.130·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6490689064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6490689064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 3 | \( 1 + 0.113T + 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 0.0940T + 19T^{2} \) |
| 23 | \( 1 + 0.210T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 - 0.289T + 53T^{2} \) |
| 59 | \( 1 - 6.22T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.86T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 4.00T + 97T^{2} \) |
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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
−9.010728642061042456138191006032, −8.724094157273534950467533635200, −7.87982751997939832982274440390, −6.70274528252664356229016204013, −6.38780195653008102033064122543, −5.51455197612627505948507372085, −4.00159313296219771568722718443, −3.02002562473926931782596836135, −1.87949008303198128994867748883, −0.67517952078201865958138698191,
0.67517952078201865958138698191, 1.87949008303198128994867748883, 3.02002562473926931782596836135, 4.00159313296219771568722718443, 5.51455197612627505948507372085, 6.38780195653008102033064122543, 6.70274528252664356229016204013, 7.87982751997939832982274440390, 8.724094157273534950467533635200, 9.010728642061042456138191006032
