L(s) = 1 | + (−0.5 − 0.866i)5-s + (2 − 1.73i)7-s + (−0.5 + 0.866i)11-s − 3·13-s + (−1 + 1.73i)17-s + (2.5 + 4.33i)19-s + (3.5 + 6.06i)23-s + (−0.499 + 0.866i)25-s + 6·29-s + (−2 + 3.46i)31-s + (−2.5 − 0.866i)35-s + (2.5 + 4.33i)37-s + 5·41-s + 6·43-s + (−4.5 − 7.79i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (0.755 − 0.654i)7-s + (−0.150 + 0.261i)11-s − 0.832·13-s + (−0.242 + 0.420i)17-s + (0.573 + 0.993i)19-s + (0.729 + 1.26i)23-s + (−0.0999 + 0.173i)25-s + 1.11·29-s + (−0.359 + 0.622i)31-s + (−0.422 − 0.146i)35-s + (0.410 + 0.711i)37-s + 0.780·41-s + 0.914·43-s + (−0.656 − 1.13i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.799051544\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.799051544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.791946621318299684379746035300, −8.119280721857280814751684692802, −7.44288011746222660204987632086, −6.85275071721364271345223966334, −5.60254243951814978616321989377, −4.98385438459882192804490140952, −4.20051254021827887462923809206, −3.30838618782954384664330983332, −2.00001611572601599166939260611, −0.980643232192021474666304123604,
0.77128339469304737573022260051, 2.44757560121531930078913341989, 2.79531972754926024125097723619, 4.28800219002226862995000045631, 4.89492322202028181960637548424, 5.72948531995675413185595193614, 6.67278328980863737766427790831, 7.42581049719476043395770205293, 8.068892527288288142877442889650, 8.972297904479883238945647392118