L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.758 + 1.83i)5-s + (1.22 + 1.22i)7-s + 0.999i·8-s + (0.707 − 0.707i)9-s + (−0.258 − 1.96i)10-s + (−1.67 − 0.448i)14-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (0.0999 + 0.241i)19-s + (1.20 + 1.57i)20-s + (−2.07 − 2.07i)25-s + (1.67 − 0.448i)28-s − 31-s + (0.866 + 0.499i)32-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.758 + 1.83i)5-s + (1.22 + 1.22i)7-s + 0.999i·8-s + (0.707 − 0.707i)9-s + (−0.258 − 1.96i)10-s + (−1.67 − 0.448i)14-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)18-s + (0.0999 + 0.241i)19-s + (1.20 + 1.57i)20-s + (−2.07 − 2.07i)25-s + (1.67 − 0.448i)28-s − 31-s + (0.866 + 0.499i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6973545867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6973545867\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (0.758 - 1.83i)T + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.0999 - 0.241i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.465 - 1.12i)T + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - 0.517T + T^{2} \) |
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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
−10.55588775676680007124069280189, −9.537338790240994997720368286429, −8.741355726542380130986317248274, −7.71479296738984325922802142679, −7.37461264431877246396050041440, −6.37832357658990541973374154441, −5.67087732030551227696954513437, −4.28362121878594636236645316073, −2.93644172990594267082840802159, −1.89177263019030237177331150670,
0.976204087452242233620768990051, 1.79497060077479748981149838072, 3.86305746813030228004883090264, 4.44130046117211943440057074995, 5.21294412685319357868704066648, 7.10323761467315787148918649275, 7.78657181047363771603768344976, 8.165207636041977937034912482472, 9.020767084914561989344717709780, 9.871663627289029535578630322232