L(s) = 1 | + 2-s + (0.396 + 1.68i)3-s + 4-s − 2.37i·5-s + (0.396 + 1.68i)6-s + (0.792 + 2.52i)7-s + 8-s + (−2.68 + 1.33i)9-s − 2.37i·10-s + 5.74·11-s + (0.396 + 1.68i)12-s + (−3.46 + i)13-s + (0.792 + 2.52i)14-s + (4 − 0.939i)15-s + 16-s − 2.52·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.228 + 0.973i)3-s + 0.5·4-s − 1.06i·5-s + (0.161 + 0.688i)6-s + (0.299 + 0.954i)7-s + 0.353·8-s + (−0.895 + 0.445i)9-s − 0.750i·10-s + 1.73·11-s + (0.114 + 0.486i)12-s + (−0.960 + 0.277i)13-s + (0.211 + 0.674i)14-s + (1.03 − 0.242i)15-s + 0.250·16-s − 0.612·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.27915 + 0.985061i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27915 + 0.985061i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.396 - 1.68i)T \) |
| 7 | \( 1 + (-0.792 - 2.52i)T \) |
| 13 | \( 1 + (3.46 - i)T \) |
good | 5 | \( 1 + 2.37iT - 5T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 - 7.57T + 19T^{2} \) |
| 23 | \( 1 - 6.78iT - 23T^{2} \) |
| 29 | \( 1 + 7.57iT - 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 + 4.90iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 4.62iT - 47T^{2} \) |
| 53 | \( 1 + 3.16iT - 53T^{2} \) |
| 59 | \( 1 + 8.74iT - 59T^{2} \) |
| 61 | \( 1 - 3.74iT - 61T^{2} \) |
| 67 | \( 1 + 2.67iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 + 9.37T + 79T^{2} \) |
| 83 | \( 1 + 4.74iT - 83T^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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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
−11.38628127637904435169483329110, −9.543663759456154561807352476266, −9.456179020237659057703593739982, −8.496458280633405105915488532786, −7.31110668599253132701773203293, −5.87341830582384439038235395713, −5.16116917247425565542594076053, −4.37973124516603654427053034982, −3.36138237782426975045777791441, −1.84292895993510538927001192635,
1.38180897512599059992427327185, 2.83984808060290603046460499705, 3.72681711696237823516420305812, 5.06045239346437786960517159697, 6.51055142053595585407815700676, 6.92390324824805067431763907155, 7.52716362163611091759966349044, 8.805163147234686117647759669919, 9.973757669124735798167233912716, 10.97709274171450805468954010152