| L(s) = 1 | − 0.386·2-s − 3-s − 1.85·4-s − 2.37·5-s + 0.386·6-s + 7-s + 1.48·8-s + 9-s + 0.917·10-s + 0.691·11-s + 1.85·12-s + 4.94·13-s − 0.386·14-s + 2.37·15-s + 3.12·16-s − 2.56·17-s − 0.386·18-s − 2.87·19-s + 4.39·20-s − 21-s − 0.267·22-s − 5.51·23-s − 1.48·24-s + 0.637·25-s − 1.90·26-s − 27-s − 1.85·28-s + ⋯ |
| L(s) = 1 | − 0.273·2-s − 0.577·3-s − 0.925·4-s − 1.06·5-s + 0.157·6-s + 0.377·7-s + 0.526·8-s + 0.333·9-s + 0.290·10-s + 0.208·11-s + 0.534·12-s + 1.37·13-s − 0.103·14-s + 0.613·15-s + 0.781·16-s − 0.621·17-s − 0.0910·18-s − 0.659·19-s + 0.982·20-s − 0.218·21-s − 0.0569·22-s − 1.14·23-s − 0.303·24-s + 0.127·25-s − 0.374·26-s − 0.192·27-s − 0.349·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 0.386T + 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 - 0.691T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 + 7.76T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 0.421T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 - 0.571T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 1.16T + 61T^{2} \) |
| 67 | \( 1 + 0.941T + 67T^{2} \) |
| 71 | \( 1 - 2.85T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−7.77681861635402926229250282746, −6.74863685689629921330002988418, −6.14903529685560092503471769097, −5.30828926106230994899012634595, −4.52177388645453283515441548744, −4.00429984209191984928371223493, −3.50171888616503526459808331804, −1.96862336589736092594943083938, −0.939408982515082695929465191308, 0,
0.939408982515082695929465191308, 1.96862336589736092594943083938, 3.50171888616503526459808331804, 4.00429984209191984928371223493, 4.52177388645453283515441548744, 5.30828926106230994899012634595, 6.14903529685560092503471769097, 6.74863685689629921330002988418, 7.77681861635402926229250282746
