| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s − 4·13-s − 2·14-s − 16-s − 6·17-s + 6·19-s − 20-s − 4·23-s + 25-s + 4·26-s − 2·28-s + 6·29-s + 8·31-s − 5·32-s + 6·34-s + 2·35-s − 6·37-s − 6·38-s + 3·40-s − 6·41-s + 6·43-s + 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.10·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s − 0.986·37-s − 0.973·38-s + 0.474·40-s − 0.937·41-s + 0.914·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.017793086743598694281416112863, −7.24125778469533790692442723144, −6.57006303177130665405311043825, −5.51841037900744680559944025854, −4.71826128070397951740716941416, −4.48379582514255780675005499843, −3.11551783391647075558634882317, −2.11305576069239070091241887537, −1.27112542139718822332489563723, 0,
1.27112542139718822332489563723, 2.11305576069239070091241887537, 3.11551783391647075558634882317, 4.48379582514255780675005499843, 4.71826128070397951740716941416, 5.51841037900744680559944025854, 6.57006303177130665405311043825, 7.24125778469533790692442723144, 8.017793086743598694281416112863
