L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999·10-s + (−0.866 − 1.5i)11-s + (0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + 0.999·10-s + (−0.866 − 1.5i)11-s + (0.866 + 0.499i)12-s + 0.999i·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6428271796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6428271796\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−10.24990375266783894208143728593, −9.193024435960010970810667163703, −8.276855337427240096621370564326, −7.86189964280162824692180604220, −7.04570626348119670308580300052, −6.25516660109818317283211771018, −4.15956083774867056343353984795, −3.29630953780711754438619651704, −2.67895887576816873589101813255, −0.78080496692233655192936998471,
2.00325116752297824852606799818, 3.09985963295911593686661986388, 4.62842248268027809634048145414, 5.22899022101729665070302856866, 6.72111745596697213266417962639, 7.53749228545587394862456934963, 8.293252034786180950949521422061, 8.853721284362381483867225216820, 9.734211731518223675409328245262, 10.22590914524953154160621070651