L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)18-s + i·19-s + (−1.67 + 0.448i)22-s + (−0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.499 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.5 − 0.866i)11-s + (0.707 − 0.707i)12-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)18-s + i·19-s + (−1.67 + 0.448i)22-s + (−0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8523892710\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8523892710\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.167578102652838360655133759835, −8.452349886923554611610272356918, −7.84498758239077008481581751899, −6.98986364125442368208471471597, −6.37445214279210025492399754986, −5.59463210121351114879283536307, −3.78645836639652196422692891349, −3.12345269070405923079614797227, −1.88012208742540114955126355831, −0.952510630613226754027963689456,
1.49800344446114196821943808396, 2.73558867257708728020251244636, 3.84218355722644797916576364930, 4.76005647335261544763765262750, 5.78540996758813203880582903588, 6.61694521044759064324016681333, 7.43062319367955670068882388374, 8.338430484982705748558783926813, 9.057603646284000994657965939971, 9.524224915738243248651389414841