L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.499i)6-s + (−0.5 + 0.866i)7-s − i·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s + i·11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.499i)14-s + (0.866 − 0.499i)15-s + (0.5 − 0.866i)16-s − i·17-s − 0.999i·18-s + (0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.499i)6-s + (−0.5 + 0.866i)7-s − i·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s + i·11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.499i)14-s + (0.866 − 0.499i)15-s + (0.5 − 0.866i)16-s − i·17-s − 0.999i·18-s + (0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8737431936\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8737431936\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 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
−9.163554470528310815585079771380, −8.154914227244949135835321941430, −7.19439360972823559535037813626, −6.51978781312262476474856390335, −5.45970309683497098419536603793, −5.12297451356591047288360220366, −4.47320360553104602686972541932, −3.62021712429457352504773802963, −2.72278983682119099063468410852, −0.48135976976241360602414322786,
1.37720298372323427136629263249, 2.79297315653459468345230347091, 3.41758204157106283801856732589, 4.27902557138296174432113779386, 5.00568009667819703071630147223, 6.31261346320391692740356884802, 6.49918032043493680527505861284, 7.72662212100072625809239269001, 7.990129098618448704167724439144, 8.901834095451085869614143988875