L(s) = 1 | + (0.5 − 0.866i)3-s + (0.292 + 0.507i)5-s + (−0.499 − 0.866i)9-s + (−2.41 + 4.18i)11-s − 4.24·13-s + 0.585·15-s + (2.29 − 3.97i)17-s + (−0.585 − 1.01i)19-s + (0.414 + 0.717i)23-s + (2.32 − 4.03i)25-s − 0.999·27-s − 2.82·29-s + (−1.41 + 2.44i)31-s + (2.41 + 4.18i)33-s + (−4.82 − 8.36i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.130 + 0.226i)5-s + (−0.166 − 0.288i)9-s + (−0.727 + 1.26i)11-s − 1.17·13-s + 0.151·15-s + (0.556 − 0.963i)17-s + (−0.134 − 0.232i)19-s + (0.0863 + 0.149i)23-s + (0.465 − 0.806i)25-s − 0.192·27-s − 0.525·29-s + (−0.254 + 0.439i)31-s + (0.420 + 0.727i)33-s + (−0.793 − 1.37i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5203077489\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5203077489\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.292 - 0.507i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + (-2.29 + 3.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.585 + 1.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.414 - 0.717i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + (1.41 - 2.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + (6.24 + 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.24 + 7.34i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.53 + 2.65i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.65 - 9.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + (-8.12 + 14.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.17 - 2.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-7.12 - 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−8.580900096405424839479940723401, −7.67663107209454391558905503129, −7.16953636970644580354936660235, −6.61327378362840887320280328035, −5.22166565558260161416138467365, −4.92681701256126270923876143523, −3.56691575099585083452908055408, −2.55332808149320120188506239331, −1.89781058372067096481678535076, −0.15613905863604332750944544085,
1.55942041022311621935613883421, 2.86574559246936318204052415230, 3.48152635749259623674752739242, 4.63588650196639138261257921762, 5.34371931563698391740734838255, 6.03716766738092625966123426370, 7.11457136171988683355072387534, 8.041505306385572563717992696595, 8.461750730556754326841350733631, 9.350806380326855271174977005258