L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.965 − 0.258i)3-s + (0.499 + 0.866i)4-s + (0.707 − 1.22i)5-s + (0.707 + 0.707i)6-s − 0.999i·8-s + (0.866 + 0.499i)9-s + (−1.22 + 0.707i)10-s + (−0.258 − 0.965i)12-s − 1.41i·13-s + (−1 + 0.999i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (1.22 + 0.707i)19-s + 1.41·20-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.965 − 0.258i)3-s + (0.499 + 0.866i)4-s + (0.707 − 1.22i)5-s + (0.707 + 0.707i)6-s − 0.999i·8-s + (0.866 + 0.499i)9-s + (−1.22 + 0.707i)10-s + (−0.258 − 0.965i)12-s − 1.41i·13-s + (−1 + 0.999i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (1.22 + 0.707i)19-s + 1.41·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5850182093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5850182093\) |
\(L(1)\) |
|
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 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.809947254820757337652285742733, −9.095323499424937779058084945253, −8.053015105798031298952147086067, −7.54001545598417861640102004283, −6.29002576914777133318604634515, −5.52925115271618508749269518832, −4.73571867274999228944556266721, −3.31754134115089003049067924610, −1.79719917928050684256294939914, −0.852383441319613939916401809882,
1.54631705780990861602848592824, 2.83294550777419548811477524389, 4.42395707554104764263749560416, 5.52735469781769772655096910686, 6.19992450362334696412961950203, 6.98377807844384531771129251143, 7.32130315856007154420644091577, 8.831932477079463479522825277170, 9.617176088532957413470881588366, 10.06732435428720828756640253756