L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.499i)6-s + i·7-s − 8-s + (0.866 + 0.499i)10-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)14-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s − i·19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.866 + 0.499i)24-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.499i)6-s + i·7-s − 8-s + (0.866 + 0.499i)10-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)14-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s − i·19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.866 + 0.499i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5562770662\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5562770662\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 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 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
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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
−13.11171005371797173645660263897, −11.93830801040101983817239437555, −11.50491176724106319037079544844, −10.17226644292946765125862866461, −9.180058689512610881312014020254, −8.159577564624528159149562240696, −7.19808119128069739656818853077, −5.49083214680502202454568705016, −3.20872019042703496496577222133, −2.33705050726053221832002406374,
3.33065637603436017905043258494, 4.42935228777675251968558759391, 6.40277395728024042605463354872, 7.75107999141018477134556915583, 8.142770115043182775628926715441, 9.285612671148327135423610248964, 10.25281064866352122952133148856, 11.85638185382237433096216740178, 12.60208685449650505161038028171, 14.35536971955271907032549400805