L(s) = 1 | + (0.965 − 0.258i)2-s + (0.382 + 0.923i)3-s + (0.866 − 0.499i)4-s + (0.923 + 0.382i)5-s + (0.608 + 0.793i)6-s + (−0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.991 + 0.130i)10-s + (0.793 + 0.608i)12-s + (0.198 − 0.739i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.58i·19-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.382 + 0.923i)3-s + (0.866 − 0.499i)4-s + (0.923 + 0.382i)5-s + (0.608 + 0.793i)6-s + (−0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.991 + 0.130i)10-s + (0.793 + 0.608i)12-s + (0.198 − 0.739i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.58i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.759033012\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.759033012\) |
\(L(1)\) |
|
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 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - 1.58iT - T^{2} \) |
| 23 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.608 - 1.05i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.599973364413482373694118686352, −8.234068492569865984491234665058, −7.60845751608285594422377210766, −6.44788451374625227211039363537, −5.90604759657361563603167798337, −5.15582339756123122295707219201, −4.11142847423277025343156253657, −3.60432588948813795809454946676, −2.70391841363885151906762131545, −1.67487444112240924189065883340,
1.72708871212233046978396059679, 2.34016526370575843952182991497, 3.18384109081006450428154271446, 4.40995438251819002013899484879, 5.34852498017430185948816755015, 6.11054977914459474519098420160, 6.46276845931942396007138417937, 7.37450574839785687753055005500, 8.240959921676509247432576684700, 8.988964743328573833239708846650