| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s − 4·11-s + 3·13-s + 16-s − 4·17-s + 3·20-s + 4·22-s − 23-s + 4·25-s − 3·26-s − 3·29-s + 6·31-s − 32-s + 4·34-s − 9·37-s − 3·40-s + 9·41-s − 3·43-s − 4·44-s + 46-s − 7·47-s − 4·50-s + 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s − 1.20·11-s + 0.832·13-s + 1/4·16-s − 0.970·17-s + 0.670·20-s + 0.852·22-s − 0.208·23-s + 4/5·25-s − 0.588·26-s − 0.557·29-s + 1.07·31-s − 0.176·32-s + 0.685·34-s − 1.47·37-s − 0.474·40-s + 1.40·41-s − 0.457·43-s − 0.603·44-s + 0.147·46-s − 1.02·47-s − 0.565·50-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.85246959404697, −15.59866799490326, −14.94369937618846, −14.18713553571539, −13.51423391389153, −13.42884429511091, −12.70511613260107, −12.10134207825107, −11.22103557785677, −10.81586660082728, −10.37037725012787, −9.732715398116441, −9.348219294207286, −8.605619995061618, −8.210997821878132, −7.503457153982083, −6.639812128064931, −6.353623841616030, −5.542062547063662, −5.169973208802493, −4.210264582657301, −3.257906533092283, −2.450812715168408, −2.003426418152265, −1.148663405317282, 0,
1.148663405317282, 2.003426418152265, 2.450812715168408, 3.257906533092283, 4.210264582657301, 5.169973208802493, 5.542062547063662, 6.353623841616030, 6.639812128064931, 7.503457153982083, 8.210997821878132, 8.605619995061618, 9.348219294207286, 9.732715398116441, 10.37037725012787, 10.81586660082728, 11.22103557785677, 12.10134207825107, 12.70511613260107, 13.42884429511091, 13.51423391389153, 14.18713553571539, 14.94369937618846, 15.59866799490326, 15.85246959404697
