| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.104 − 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−4.34 + 0.923i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.104 + 0.994i)10-s + (0.280 + 0.311i)11-s + (−0.913 − 0.406i)12-s + (−4.57 + 2.03i)13-s + (−2.97 + 3.30i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.519 + 0.577i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.0603 − 0.574i)3-s + (0.154 − 0.475i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−1.64 + 0.349i)7-s + (−0.109 − 0.336i)8-s + (−0.326 − 0.0693i)9-s + (0.0330 + 0.314i)10-s + (0.0846 + 0.0940i)11-s + (−0.263 − 0.117i)12-s + (−1.26 + 0.564i)13-s + (−0.794 + 0.882i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.126 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.277012 + 0.348975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277012 + 0.348975i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (4.06 - 3.80i)T \) |
| good | 7 | \( 1 + (4.34 - 0.923i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.280 - 0.311i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (4.57 - 2.03i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (0.519 - 0.577i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.80 - 2.13i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 5.33i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.73 - 2.71i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (3.38 + 5.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.00274 + 0.0260i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (6.04 + 2.68i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (6.32 + 4.59i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.05 + 0.436i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.124 + 1.18i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + (3.48 - 6.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (14.2 + 3.03i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-7.94 - 8.82i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (4.29 - 4.77i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.629 + 5.98i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-3.59 + 11.0i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.76 + 11.5i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.18770591845061776570572913566, −9.635985759215606239922261441544, −8.867483573645333009661578551507, −7.24629520350366974156412888949, −7.07906203080696520547707526943, −5.96413429342581085566408158774, −5.16886167742860116838478110638, −3.63668486298262220476899863118, −3.07475303311902203157205537672, −1.88189819983373644725786204596,
0.15843383399572031298663119622, 2.77673387271845744826224550431, 3.46361085792409901118951461506, 4.55178432308815176611011114334, 5.33527217824459757663258649520, 6.36876237176470445910760072780, 7.16038335476886176926835390363, 8.008862371400722410191161042606, 9.224966917202299636867302233845, 9.675643009716429586167398949924
