L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 + 0.499i)4-s − 6-s + (−0.448 + 1.67i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + 1.73i·21-s + (0.965 − 0.258i)23-s + (−0.866 − 0.5i)24-s + (0.707 − 0.707i)27-s + (−1.22 + 1.22i)28-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 + 0.499i)4-s − 6-s + (−0.448 + 1.67i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)12-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + 1.73i·21-s + (0.965 − 0.258i)23-s + (−0.866 − 0.5i)24-s + (0.707 − 0.707i)27-s + (−1.22 + 1.22i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8906747879\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8906747879\) |
\(L(1)\) |
|
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 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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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
−10.02373553589881799172647810226, −9.313240013517136702481182114087, −8.683538056586852223441645113623, −8.256964445545631621083703820099, −7.03731205404717042542235628203, −6.43757408701832304355870789163, −5.15211957773018632456610951007, −3.40272243571631751407274821507, −2.74245510956883595084554688320, −1.72323239918466992563599952257,
1.24402369658114228747273641280, 2.76288106049437768724421681802, 3.78063076077313231775873546489, 4.88309281972390220065856734651, 6.46849415199553343750967105821, 7.10330265727257533579765076224, 7.86525121906230627827701684722, 8.529778731429379337881364114988, 9.640269814948362681881119229610, 9.978195540704872921247805434253