| L(s) = 1 | − 2.23i·2-s − 3.00·4-s + 2.23i·5-s + 2.23i·8-s + 5.00·10-s − 0.999·16-s + 4.47i·17-s − 4·19-s − 6.70i·20-s − 8.94i·23-s − 5.00·25-s + 8·31-s + 6.70i·32-s + 10.0·34-s + 8.94i·38-s + ⋯ |
| L(s) = 1 | − 1.58i·2-s − 1.50·4-s + 0.999i·5-s + 0.790i·8-s + 1.58·10-s − 0.249·16-s + 1.08i·17-s − 0.917·19-s − 1.50i·20-s − 1.86i·23-s − 1.00·25-s + 1.43·31-s + 1.18i·32-s + 1.71·34-s + 1.45i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.541744 - 0.541744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.541744 - 0.541744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 2 | \( 1 + 2.23iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8.94iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−15.31086366215355683397844908499, −14.18964599311887806872946724336, −12.96574043565923018826779476430, −11.92843380880703408712681539674, −10.71802248997556069743590768513, −10.18166894925267983021659200493, −8.529513714199935388900375216623, −6.52566376180141557577636495203, −4.10424053282794606423576741067, −2.49439517148849264400889437608,
4.63076666346438307078059690242, 5.84873109781954041807523169102, 7.36447877954871795658687378464, 8.506199327349381512983870230050, 9.570850536528072839705758819793, 11.66876266699699984465493727952, 13.15723379959748243640876161274, 14.03829554572186610771548305855, 15.39860978254693558799803182116, 16.04425621665513637432335881101
