L(s) = 1 | + 1.41·2-s + (1 − 1.41i)3-s − i·5-s + (1.41 − 2.00i)6-s − i·7-s − 2.82·8-s + (−1.00 − 2.82i)9-s − 1.41i·10-s + (−3 − 1.41i)11-s + 2.24i·13-s − 1.41i·14-s + (−1.41 − i)15-s − 4.00·16-s − 5.82·17-s + (−1.41 − 4.00i)18-s − 3.24i·19-s + ⋯ |
L(s) = 1 | + 1.00·2-s + (0.577 − 0.816i)3-s − 0.447i·5-s + (0.577 − 0.816i)6-s − 0.377i·7-s − 0.999·8-s + (−0.333 − 0.942i)9-s − 0.447i·10-s + (−0.904 − 0.426i)11-s + 0.621i·13-s − 0.377i·14-s + (−0.365 − 0.258i)15-s − 1.00·16-s − 1.41·17-s + (−0.333 − 0.942i)18-s − 0.743i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.892414867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.892414867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 13 | \( 1 - 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 3.24iT - 19T^{2} \) |
| 23 | \( 1 + 1.24iT - 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 3.24iT - 43T^{2} \) |
| 47 | \( 1 + 1.07iT - 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + 12.8iT - 59T^{2} \) |
| 61 | \( 1 - 9.24iT - 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 1.75iT - 71T^{2} \) |
| 73 | \( 1 + 6.48iT - 73T^{2} \) |
| 79 | \( 1 + 6.24iT - 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 16.4iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.139758022301646199359626877677, −8.650414389463137057219876824771, −7.83861477618943217929226722440, −6.68341426209116621916871477674, −6.19853898811194708671325459513, −4.91179356007377517280699769106, −4.33492508349302071643485766775, −3.13364371502824762233945121972, −2.28351808249928811457742565149, −0.51968838239840333391418286012,
2.49604253961629437896164921632, 3.00108665063701512194098575358, 4.19384811111398399880030679746, 4.77517801951387177258266818107, 5.66180039340876746857808495419, 6.52321052831688718640739036650, 7.86203151545594919909184161528, 8.475996808158554914037102510333, 9.411180763752445079242197060346, 10.13192991536889851883323599606