L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s + 2·9-s + 6·11-s − i·12-s + i·13-s − 14-s + 16-s + 3i·17-s + 2i·18-s − 2·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s − 0.353i·8-s + 0.666·9-s + 1.80·11-s − 0.288i·12-s + 0.277i·13-s − 0.267·14-s + 0.250·16-s + 0.727i·17-s + 0.471i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.818042 + 1.32362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.818042 + 1.32362i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.70098877620937336582291918831, −9.680829457374314976377778210585, −9.148684495084243472568470252189, −8.346945031274088038464988465007, −7.10635703991053365887461595420, −6.46195641362797225618709574511, −5.44285957357035627025137933025, −4.25897165368108855897184064932, −3.70206482423164327056373533438, −1.63692155797874181734667240244,
0.972966383262139416307914746253, 2.08209392662962531596997767351, 3.66515659246915443880282740978, 4.35717487180676069188662703487, 5.76685307269462794444329466436, 6.88497879612270532939865279275, 7.48759135038314238106148190187, 8.774127922934510844182503440453, 9.442884282294208701163240007034, 10.29638044181837094320659440859