L(s) = 1 | + (0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)11-s − 0.999i·15-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + i·27-s + (−0.866 + 0.5i)31-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.866 − 0.5i)47-s + (0.866 + 0.499i)51-s + (0.5 + 0.866i)53-s − 0.999i·55-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)11-s − 0.999i·15-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)23-s + i·27-s + (−0.866 + 0.5i)31-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.866 − 0.5i)47-s + (0.866 + 0.499i)51-s + (0.5 + 0.866i)53-s − 0.999i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.601338274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.601338274\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.100378494918809707838414240606, −8.799180476661782570439087289467, −8.123269742053224951970399587175, −7.23364939266347362695868576329, −6.23394325079814120440714645491, −5.50792528779483325711728324327, −4.39176794211603451297949411961, −3.45747944265692077983171343238, −2.23764522966363915183976871933, −1.37932746312004637541757497025,
1.86529987310190334708636626818, 2.84643444050875121721830909544, 3.67762881048408347299678897477, 4.49873285644506070840628310648, 5.81867498413783790780442073604, 6.51138610755418176410632535995, 7.36299469148400393871944311330, 8.263540235536589025580688392462, 9.060354555844478392314609000656, 9.923477613604870105420086083679