L(s) = 1 | + (2.59 + 1.5i)3-s + (2.59 − 0.5i)7-s + (3 + 5.19i)9-s + (1 − 1.73i)11-s − 6i·13-s + (1.73 + i)17-s + (7.5 + 2.59i)21-s + (−7.79 + 4.5i)23-s + 9i·27-s − 3·29-s + (−1 + 1.73i)31-s + (5.19 − 3i)33-s + (−6.92 + 4i)37-s + (9 − 15.5i)39-s + 5·41-s + ⋯ |
L(s) = 1 | + (1.49 + 0.866i)3-s + (0.981 − 0.188i)7-s + (1 + 1.73i)9-s + (0.301 − 0.522i)11-s − 1.66i·13-s + (0.420 + 0.242i)17-s + (1.63 + 0.566i)21-s + (−1.62 + 0.938i)23-s + 1.73i·27-s − 0.557·29-s + (−0.179 + 0.311i)31-s + (0.904 − 0.522i)33-s + (−1.13 + 0.657i)37-s + (1.44 − 2.49i)39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.58992 + 0.787553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.58992 + 0.787553i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.59 + 0.5i)T \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.79 - 4.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.46 + 2i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 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
−10.34880614579111213119087344008, −9.675723734676498759228744250302, −8.706178860224578912191709691762, −7.992600297852710622843356644228, −7.63793255811454605600635851568, −5.84711342882042746286338130168, −4.87661561592478920467818965468, −3.78097153947895385409843541218, −3.10169379256012402887933854753, −1.72196293071766099245710907694,
1.70756452577681252597114183710, 2.21874116559189075748288815535, 3.72783207606035734724301947734, 4.59818711338232673591791562115, 6.18295792692781301525212610270, 7.17221391813796075269955755591, 7.76098436563763344394501535172, 8.659146693144733000392518224338, 9.175052635047281800735947634756, 10.11512271689173922218485387170