L(s) = 1 | + (0.591 − 1.02i)3-s + (−0.5 − 0.866i)5-s + (0.799 + 1.38i)9-s + (−0.115 + 0.199i)11-s − 2.27·13-s − 1.18·15-s + (−3.26 + 5.65i)17-s + (0.130 + 0.225i)19-s + (4.43 + 7.68i)23-s + (−0.499 + 0.866i)25-s + 5.44·27-s + 5.42·29-s + (−2.43 + 4.21i)31-s + (0.136 + 0.236i)33-s + (0.577 + 0.999i)37-s + ⋯ |
L(s) = 1 | + (0.341 − 0.591i)3-s + (−0.223 − 0.387i)5-s + (0.266 + 0.461i)9-s + (−0.0347 + 0.0601i)11-s − 0.630·13-s − 0.305·15-s + (−0.792 + 1.37i)17-s + (0.0298 + 0.0516i)19-s + (0.924 + 1.60i)23-s + (−0.0999 + 0.173i)25-s + 1.04·27-s + 1.00·29-s + (−0.437 + 0.757i)31-s + (0.0237 + 0.0410i)33-s + (0.0948 + 0.164i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.651596605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651596605\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.591 + 1.02i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.115 - 0.199i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + (3.26 - 5.65i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.130 - 0.225i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.43 - 7.68i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + (2.43 - 4.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.577 - 0.999i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + (0.461 + 0.800i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.06 + 1.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 4.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.44 + 7.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.07 - 7.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.06T + 71T^{2} \) |
| 73 | \( 1 + (3.00 - 5.20i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0564 + 0.0977i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.72T + 83T^{2} \) |
| 89 | \( 1 + (7.95 + 13.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−9.045160694690561879566394469717, −8.455755797820321938135396729351, −7.63559653721943305931582015479, −7.12142939031795411476570763768, −6.17134123496322239702321641223, −5.14126820891579990575417153537, −4.41312197182465915905769222106, −3.32191507265927017559038998607, −2.18576722477866530627153460138, −1.27082140780631201977324633783,
0.61455346819391059546167187309, 2.49711138393531190523432893974, 3.07700437535469134114421049514, 4.36318901009754061903650830305, 4.68469793957438166439736085895, 5.98599537303378742494772068180, 6.90689360320164515778824053697, 7.38628901180959945496195874473, 8.541340825610930879254132427090, 9.145208766114130647147020354177