L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.258 + 0.965i)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 + (−0.965 + 0.258i)12-s − 1.41i·13-s + (−0.999 − i)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 + (−0.258 + 0.965i)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 + (−0.965 + 0.258i)12-s − 1.41i·13-s + (−0.999 − i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (1.22 + 0.707i)19-s − 1.41·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.338785930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.338785930\) |
\(L(1)\) |
|
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 | \( 1 + (0.258 - 0.965i)T \) |
| 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
−10.53892499623865089031631795134, −9.722307749842525116002084424638, −8.422760341999980361273456003908, −7.71723390110773433602622240891, −6.95932560469646087587398797094, −5.91366147022336742530745912288, −5.33359548087387616560037682059, −4.20038622574382325382002568657, −3.34668286858089698264405092974, −2.87495995552795226819473969317,
0.981814426172127204849456122728, 2.07052366309325228295487796832, 3.42876115059211229191791334976, 4.58302899635765665440067393240, 5.11780773223273775934669119598, 6.17560787670356677560915014167, 7.03144935107008582638003360448, 7.79547462097712172457471430239, 8.883376111771230452269644610669, 9.490112048889612040582007586927