L(s) = 1 | + (−0.866 + 0.5i)3-s − i·4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (−0.5 + 0.133i)19-s + (−0.499 + 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 − 0.866i)28-s + (−1.36 + 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s − i·4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s − 16-s + (−0.5 + 0.133i)19-s + (−0.499 + 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 − 0.866i)28-s + (−1.36 + 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6415837040\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6415837040\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + iT^{2} \) |
| 5 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−11.56638594159037571451617487971, −11.17940899166395310978364890070, −10.31026669631553263824514617642, −9.523346298909165842596630898267, −8.308660176718113188897170199922, −6.82387818687788767839684934779, −5.97423186250830402451939335800, −4.93757266341048901716305037813, −4.03834894780255131311491278359, −1.50214803518883393025586550229,
2.07231612362859248253041835300, 3.87042298282541794138872455744, 5.17358557078102877837631555532, 6.19970708999153305762628923764, 7.46453818625146142633300934425, 8.107211141571774721641999490708, 9.151878976193034306692145014547, 10.77008989429976638105072027551, 11.34984707967465724632257761781, 12.14283645568478792849058137253