L(s) = 1 | + (1.5 + 2.59i)5-s + (−0.5 − 2.59i)7-s + (2.5 − 4.33i)11-s + 2·13-s + (1 − 1.73i)17-s + (−3 − 5.19i)19-s + (1 + 1.73i)23-s + (−2 + 3.46i)25-s − 29-s + (−0.5 + 0.866i)31-s + (6 − 5.19i)35-s + (−5 − 8.66i)37-s − 4·41-s − 4·43-s + (−4 − 6.92i)47-s + ⋯ |
L(s) = 1 | + (0.670 + 1.16i)5-s + (−0.188 − 0.981i)7-s + (0.753 − 1.30i)11-s + 0.554·13-s + (0.242 − 0.420i)17-s + (−0.688 − 1.19i)19-s + (0.208 + 0.361i)23-s + (−0.400 + 0.692i)25-s − 0.185·29-s + (−0.0898 + 0.155i)31-s + (1.01 − 0.878i)35-s + (−0.821 − 1.42i)37-s − 0.624·41-s − 0.609·43-s + (−0.583 − 1.01i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.886629004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886629004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 + 4.33i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.5 + 11.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 19T + 97T^{2} \) |
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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
−9.045123125375369896602600424744, −8.353570762400622730150495332305, −7.13101119117500357847415777651, −6.76616668862170428917346915030, −6.06752177642340291844027681224, −5.10967807210319452549296861953, −3.74719883940556922878294168727, −3.32034840118606920397202253026, −2.12160047436661961584688419465, −0.69586354198587284954155924303,
1.44327569818942655727504395071, 2.03755895101904677500674640748, 3.50294567148059797261250200292, 4.54431664561376854670460912610, 5.22780512147982155854766031564, 6.09894701559066043476933716329, 6.65948385872621481819399551257, 7.971922064393967368105895191161, 8.602747297073097297863471648108, 9.235369252272996513624492306330