L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.294 − 0.169i)5-s − 0.999i·6-s + (0.420 + 2.61i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.169 + 0.294i)10-s + (−0.571 − 0.330i)11-s + (0.499 + 0.866i)12-s + (0.660 + 3.54i)13-s + (−1.66 − 2.05i)14-s + 0.339i·15-s + (−0.5 − 0.866i)16-s + (3.27 − 5.66i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.131 − 0.0759i)5-s − 0.408i·6-s + (0.158 + 0.987i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.0537 + 0.0930i)10-s + (−0.172 − 0.0995i)11-s + (0.144 + 0.249i)12-s + (0.183 + 0.983i)13-s + (−0.446 − 0.548i)14-s + 0.0877i·15-s + (−0.125 − 0.216i)16-s + (0.793 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.296885 + 0.735842i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.296885 + 0.735842i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.420 - 2.61i)T \) |
| 13 | \( 1 + (-0.660 - 3.54i)T \) |
good | 5 | \( 1 + (-0.294 + 0.169i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.571 + 0.330i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.27 + 5.66i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.27 - 3.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.71 - 6.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + (-3.46 - 2i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.06 - 1.77i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.864iT - 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + (8.18 - 4.72i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.38 + 1.95i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.932 + 1.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.45 - 3.72i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.884iT - 71T^{2} \) |
| 73 | \( 1 + (-8.72 - 5.03i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.22 - 10.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.2iT - 83T^{2} \) |
| 89 | \( 1 + (-3.85 + 2.22i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−11.25612347021864638739768188436, −9.963582313934363726213939494841, −9.382848121577874140099155800034, −8.671401815333167922896752149928, −7.64417200730719357436910698735, −6.55814281653377241874415190888, −5.60510591746501061308238128253, −4.86918619435096282797047956465, −3.29795461025429757316418094837, −1.75241520128136213562531316942,
0.58191891545682600637432195264, 2.04246026006739435575504231147, 3.47744170240539154964307239828, 4.72489286943084607449331449332, 6.14284066759589232986473249716, 6.89446589311227127453879897089, 8.031334138575829540756194908187, 8.401041646166225740007442714833, 9.865811839898719707422329091994, 10.61585523491372950814731313085