L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.669 + 0.743i)3-s + (−0.207 + 0.978i)5-s + (0.207 − 0.978i)6-s + (0.5 + 0.866i)7-s + (−0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.913 − 0.406i)10-s + (0.587 + 0.809i)11-s + (0.0646 − 0.614i)13-s + (0.406 − 0.913i)14-s + (−0.866 + 0.5i)15-s + (0.809 + 0.587i)16-s + (0.866 − 0.5i)18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.669 + 0.743i)3-s + (−0.207 + 0.978i)5-s + (0.207 − 0.978i)6-s + (0.5 + 0.866i)7-s + (−0.951 + 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.913 − 0.406i)10-s + (0.587 + 0.809i)11-s + (0.0646 − 0.614i)13-s + (0.406 − 0.913i)14-s + (−0.866 + 0.5i)15-s + (0.809 + 0.587i)16-s + (0.866 − 0.5i)18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.126387539\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126387539\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.207 - 0.978i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.0646 + 0.614i)T + (-0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 19 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 23 | \( 1 + (-0.614 + 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 29 | \( 1 + (0.336 + 1.58i)T + (-0.913 + 0.406i)T^{2} \) |
| 31 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 37 | \( 1 + (0.564 - 0.251i)T + (0.669 - 0.743i)T^{2} \) |
| 41 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 59 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.669 + 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 71 | \( 1 + (-0.336 - 1.58i)T + (-0.913 + 0.406i)T^{2} \) |
| 73 | \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \) |
| 79 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−9.146332368800405371602637063155, −8.385518064598052233791602227311, −7.84630639907359874165448657978, −6.76870354438602119934068108450, −5.91522311322425245450045775659, −5.03218922453517560308772708770, −4.07743657320690327816714234560, −3.04225315744056123357729221955, −2.54326026951245003998495932975, −1.66686415146711723949913708402,
0.78494777105636345328673499092, 1.75026017408925703910355054778, 3.28084576534820171621709390490, 3.86665692227660246914886673090, 4.96265836104305045214830041274, 6.00076603317048644443026975999, 6.82543963002005880138935329139, 7.29648400423636489344772659013, 8.007137730492839300280010589184, 8.717749562841451574121344788939