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
−9.235369252272996513624492306330, −8.602747297073097297863471648108, −7.971922064393967368105895191161, −6.65948385872621481819399551257, −6.09894701559066043476933716329, −5.22780512147982155854766031564, −4.54431664561376854670460912610, −3.50294567148059797261250200292, −2.03755895101904677500674640748, −1.44327569818942655727504395071,
0.69586354198587284954155924303, 2.12160047436661961584688419465, 3.32034840118606920397202253026, 3.74719883940556922878294168727, 5.10967807210319452549296861953, 6.06752177642340291844027681224, 6.76616668862170428917346915030, 7.13101119117500357847415777651, 8.353570762400622730150495332305, 9.045123125375369896602600424744