L(s) = 1 | + i·2-s − 3-s − 4-s + (0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + 9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 12-s + (−0.866 − 0.5i)13-s + 14-s + (−0.866 + 0.5i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + i·2-s − 3-s − 4-s + (0.866 − 0.5i)5-s − i·6-s − i·7-s − i·8-s + 9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 12-s + (−0.866 − 0.5i)13-s + 14-s + (−0.866 + 0.5i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6789453062\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6789453062\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−10.88824631696091301989676147952, −10.20544216970302790727177861389, −9.476967798218024169293886835911, −8.325415086944146103325691383344, −7.36914815145245358945068423064, −6.37628993849240076674372588912, −5.74929805445307728475571340276, −4.84237164066302856812954850762, −3.82083975783995934688267416312, −1.11802188838229674456774451177,
1.78811621425112818376292592246, 2.84287509212374292056016127822, 4.63075073671814750903882641780, 5.21931208877459127985061127082, 6.35515986365658450619389295055, 7.27731696251836989534443859712, 8.949983654134283142064475763572, 9.628680909830292393270021241766, 10.20618744026547122997327619732, 11.22066823969641448960092843191