L(s) = 1 | + (0.866 − 1.11i)2-s + (−0.402 − 1.68i)3-s + (−0.499 − 1.93i)4-s + (1.09 + 1.89i)5-s + (−2.23 − 1.00i)6-s + (0.451 − 2.60i)7-s + (−2.59 − 1.11i)8-s + (−2.67 + 1.35i)9-s + (3.06 + 0.417i)10-s + (1.45 + 0.837i)11-s + (−3.06 + 1.62i)12-s + 1.56i·13-s + (−2.52 − 2.76i)14-s + (2.74 − 2.60i)15-s + (−3.50 + 1.93i)16-s + (0.278 + 0.160i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.790i)2-s + (−0.232 − 0.972i)3-s + (−0.249 − 0.968i)4-s + (0.488 + 0.845i)5-s + (−0.911 − 0.412i)6-s + (0.170 − 0.985i)7-s + (−0.918 − 0.395i)8-s + (−0.892 + 0.451i)9-s + (0.967 + 0.132i)10-s + (0.437 + 0.252i)11-s + (−0.883 + 0.467i)12-s + 0.432i·13-s + (−0.674 − 0.738i)14-s + (0.709 − 0.671i)15-s + (−0.875 + 0.483i)16-s + (0.0676 + 0.0390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.805129 - 1.22409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.805129 - 1.22409i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 1.11i)T \) |
| 3 | \( 1 + (0.402 + 1.68i)T \) |
| 7 | \( 1 + (-0.451 + 2.60i)T \) |
good | 5 | \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 0.837i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.56iT - 13T^{2} \) |
| 17 | \( 1 + (-0.278 - 0.160i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.87 - 4.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.26 + 5.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.04T + 29T^{2} \) |
| 31 | \( 1 + (-7.76 - 4.48i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.946 - 0.546i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + (2.25 + 3.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.42 - 7.67i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.37 + 3.10i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.80 - 2.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.716 + 1.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.37T + 71T^{2} \) |
| 73 | \( 1 + (4.49 - 7.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.58 + 2.07i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.06iT - 83T^{2} \) |
| 89 | \( 1 + (4.57 - 2.64i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
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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
−12.33890292602215914491271023991, −11.72676795883660922296521226966, −10.54081621489477210697010440261, −10.02058964755608992813550167510, −8.280611699893354634494502367915, −6.79802292724323792221606252324, −6.24178129126313643142047262179, −4.62088705641147416558317013477, −3.01469354919199097304581013301, −1.50200081079378721373554676294,
3.10190419056834362094096367809, 4.69879145124190686781320970822, 5.39429840682996678949295190420, 6.33026927476515326943760466975, 8.152951617886395478972753449021, 8.996933441803302155834491164128, 9.749557028606913823237249995222, 11.49818532595574074912835259194, 12.07759214086840097609751531883, 13.29733686348845170718010954333