L(s) = 1 | + (−1.5 − 0.866i)3-s + (−0.5 + 2.59i)7-s + (1.5 + 2.59i)9-s + 7·13-s + (4.5 − 2.59i)19-s + (3 − 3.46i)21-s + (−2.5 + 4.33i)25-s − 5.19i·27-s + (7.5 + 4.33i)31-s + (0.5 + 0.866i)37-s + (−10.5 − 6.06i)39-s + 12.1i·43-s + (−6.5 − 2.59i)49-s − 9·57-s + (−7 − 12.1i)61-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (−0.188 + 0.981i)7-s + (0.5 + 0.866i)9-s + 1.94·13-s + (1.03 − 0.596i)19-s + (0.654 − 0.755i)21-s + (−0.5 + 0.866i)25-s − 0.999i·27-s + (1.34 + 0.777i)31-s + (0.0821 + 0.142i)37-s + (−1.68 − 0.970i)39-s + 1.84i·43-s + (−0.928 − 0.371i)49-s − 1.19·57-s + (−0.896 − 1.55i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04300 + 0.141719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04300 + 0.141719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−11.50509623888421216855033106010, −11.02778821757714182722028862004, −9.773730420857100699207285973512, −8.742674138928142622161561070810, −7.77938763746407257923605980462, −6.48815373607816860859905757546, −5.89103341909744038016857788119, −4.82736235381145090408970689515, −3.16191065780191309217603365689, −1.39779143957032138731475895823,
1.02291026116260855523232131320, 3.54874763146271656919782870647, 4.32523645769660684583815987270, 5.74001776421915450440581050507, 6.45631247953826733311180692242, 7.62660824680861555647374364162, 8.811285232859324080727195548227, 9.992587183552170040469416810071, 10.54331343173576626376388242329, 11.42700106021728381473757584862