L(s) = 1 | + (−1.02 + 0.592i)2-s + (−0.296 + 0.514i)4-s + (−1.41 + 2.45i)5-s + (−2.07 − 1.64i)7-s − 3.07i·8-s − 3.36i·10-s + (−0.136 + 0.0789i)11-s + (−3.41 − 1.97i)13-s + (3.10 + 0.460i)14-s + (1.23 + 2.13i)16-s − 4.14·17-s + 6.33i·19-s + (−0.842 − 1.45i)20-s + (0.0935 − 0.162i)22-s + (−0.472 − 0.273i)23-s + ⋯ |
L(s) = 1 | + (−0.726 + 0.419i)2-s + (−0.148 + 0.257i)4-s + (−0.634 + 1.09i)5-s + (−0.783 − 0.621i)7-s − 1.08i·8-s − 1.06i·10-s + (−0.0412 + 0.0237i)11-s + (−0.947 − 0.546i)13-s + (0.829 + 0.122i)14-s + (0.307 + 0.532i)16-s − 1.00·17-s + 1.45i·19-s + (−0.188 − 0.326i)20-s + (0.0199 − 0.0345i)22-s + (−0.0986 − 0.0569i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0330127 - 0.209935i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0330127 - 0.209935i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.07 + 1.64i)T \) |
good | 2 | \( 1 + (1.02 - 0.592i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.41 - 2.45i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.136 - 0.0789i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.41 + 1.97i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 6.33iT - 19T^{2} \) |
| 23 | \( 1 + (0.472 + 0.273i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.112 - 0.0647i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + (-1.99 + 3.45i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.28 - 5.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.33 - 7.50i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.60iT - 53T^{2} \) |
| 59 | \( 1 + (1.80 - 3.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.91 - 1.68i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.409iT - 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 + (2.16 + 3.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.22 + 5.58i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 + (2.18 - 1.26i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.98587090099710487972951500928, −12.20784219227359775837029473506, −10.81882776122704700670107447370, −10.12197131622005440970735100452, −9.117422921957328276692529852086, −7.72447440483274485922425570354, −7.29227348301532280227558817608, −6.24164527117052582538735081166, −4.13927899129749112506449740701, −3.09815308548138871674193506266,
0.22890934771112124956133219779, 2.34770525027868703003146903112, 4.41875839478143503784048271524, 5.42735654518877388975226106274, 6.98462184980413581711111744566, 8.436889242895372604677942270405, 9.096830905509749171757427398155, 9.737099233097952810292332026160, 11.09353324940274457332248433612, 11.91022749100737912366610578787