L(s) = 1 | + (−0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (−1.80 + 0.587i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.690 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s + 1.17i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 1.53i)28-s + (−1.11 − 1.53i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + 0.999·44-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)4-s + (0.309 + 0.951i)5-s + (−1.80 + 0.587i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.690 + 0.951i)17-s + (0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s + 1.17i·23-s + (−0.809 + 0.587i)25-s + (1.11 + 1.53i)28-s + (−1.11 − 1.53i)35-s + (0.309 − 0.951i)36-s − 1.17i·43-s + 0.999·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7712829715\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7712829715\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 1.17iT - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.17iT - T^{2} \) |
| 47 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T^{2} \) |
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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
−10.23938054066813213299732296273, −9.635298039313261951985931091317, −9.063946692326601879966763442585, −7.46495416281157664474304657515, −6.89080000509376905665192877996, −6.01384794047537770724602692529, −5.38092857440446642405383974307, −4.07639387084559780013526063422, −2.93456754162108800685458813777, −1.86155229764754625425459855511,
0.72978542713341857223768524333, 2.90900610716713555444577373122, 3.65237288046653406491478526048, 4.48715950418275564933690547249, 5.76535839848531155309879708630, 6.61250716204408212517590661979, 7.48408156880170114600864382361, 8.352817926728751171290607575914, 9.310499830238865128966878857218, 9.676149345411763453496921535948