L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (−0.499 − 0.866i)12-s + 13-s + (−0.866 − 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s + (0.499 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s − 0.999·6-s + (−0.866 + 0.5i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (−0.499 − 0.866i)12-s + 13-s + (−0.866 − 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s + (0.499 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6694189580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6694189580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.275906061299964732586357650984, −8.562818230838701138379792190907, −7.979623773597579266845544560381, −6.83143799083042158017572821704, −5.97503878645741320994099725266, −5.26889574161527710981887716669, −4.66652947726106630546339625491, −3.82583622271022715268581514702, −3.13138151458657835017391069080, −0.48473332227326592558812151806,
1.33865562458487585513989748969, 2.66635831856676876942035830220, 3.55253509852245714274582862659, 3.99371136546772576338000587076, 5.61224997328427317454700173524, 6.24867661999694151830304178454, 6.81094099340362001817070760710, 7.71965319445385801036402760611, 8.454985612948002637308584662552, 9.847753742189536923437035879191