L(s) = 1 | + i·2-s − 4-s + (−0.539 − 2.17i)5-s + 3.70i·7-s − i·8-s + (2.17 − 0.539i)10-s + 2·11-s − i·13-s − 3.70·14-s + 16-s + 4.78i·17-s + 0.630·19-s + (0.539 + 2.17i)20-s + 2i·22-s − 4.97i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.241 − 0.970i)5-s + 1.40i·7-s − 0.353i·8-s + (0.686 − 0.170i)10-s + 0.603·11-s − 0.277i·13-s − 0.991·14-s + 0.250·16-s + 1.16i·17-s + 0.144·19-s + (0.120 + 0.485i)20-s + 0.426i·22-s − 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.333140465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333140465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.539 + 2.17i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 - 3.70iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 4.78iT - 17T^{2} \) |
| 19 | \( 1 - 0.630T + 19T^{2} \) |
| 23 | \( 1 + 4.97iT - 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 3.41iT - 47T^{2} \) |
| 53 | \( 1 - 13.1iT - 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 7.57T + 61T^{2} \) |
| 67 | \( 1 + 0.156iT - 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 4.49iT - 83T^{2} \) |
| 89 | \( 1 + 4.73T + 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−9.628221760515261757024491875073, −9.010965602061281181373421082613, −8.370995063146929386205906112979, −7.82848302669455723858064724321, −6.38169069071020405176595021450, −5.96523827044477583015370152877, −4.93619027181143196986210446859, −4.23791003511855226928633632457, −2.84928253822083163467701975564, −1.31444206478428119964010657843,
0.65084533202457138758105610932, 2.15292252213754897274837237610, 3.44500348626596485625282796832, 3.93469436323705205695648942437, 5.03236314572215028446045782816, 6.34232493683229179961339270560, 7.25220179121081758712203756527, 7.63194613908986120504660534937, 9.000065932970017305355104473000, 9.717364180350029036238036314356