L(s) = 1 | + (1.11 − 0.642i)2-s + (2.34 − 1.86i)3-s + (−1.17 + 2.03i)4-s − 7.87i·5-s + (1.41 − 3.58i)6-s + (0.417 + 6.98i)7-s + 8.15i·8-s + (2.03 − 8.76i)9-s + (−5.06 − 8.76i)10-s + 12.3i·11-s + (1.03 + 6.96i)12-s + (−3.39 − 5.87i)13-s + (4.95 + 7.50i)14-s + (−14.6 − 18.5i)15-s + (0.548 + 0.949i)16-s + (6.19 − 3.57i)17-s + ⋯ |
L(s) = 1 | + (0.556 − 0.321i)2-s + (0.783 − 0.621i)3-s + (−0.293 + 0.508i)4-s − 1.57i·5-s + (0.236 − 0.597i)6-s + (0.0596 + 0.998i)7-s + 1.01i·8-s + (0.226 − 0.974i)9-s + (−0.506 − 0.876i)10-s + 1.12i·11-s + (0.0862 + 0.580i)12-s + (−0.261 − 0.452i)13-s + (0.353 + 0.536i)14-s + (−0.979 − 1.23i)15-s + (0.0342 + 0.0593i)16-s + (0.364 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61088 - 0.684466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61088 - 0.684466i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.34 + 1.86i)T \) |
| 7 | \( 1 + (-0.417 - 6.98i)T \) |
good | 2 | \( 1 + (-1.11 + 0.642i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + 7.87iT - 25T^{2} \) |
| 11 | \( 1 - 12.3iT - 121T^{2} \) |
| 13 | \( 1 + (3.39 + 5.87i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-6.19 + 3.57i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (15.4 - 26.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + 2.16iT - 529T^{2} \) |
| 29 | \( 1 + (8.97 + 5.17i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-10.0 + 17.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-2.46 + 4.26i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-28.0 + 16.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-25.5 + 44.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-18.5 + 10.7i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-51.0 + 29.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (89.6 + 51.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4.46 + 7.74i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (45.3 - 78.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 2.12iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-35.2 - 61.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-72.3 - 125. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (13.6 + 7.85i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-4.08 - 2.35i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (47.0 - 81.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.37027856577082783530227173837, −13.10968890060266136622835944849, −12.39241648811926505079146418740, −12.13577859606039074581581553399, −9.557807349618490460019997829645, −8.594549646496345546707962317161, −7.80477508610360020536499802944, −5.51180170546287264283975212562, −4.13136868747966340001016022547, −2.15807759223748894200042333562,
3.15505846622446309321517882608, 4.43362541168024964743387703072, 6.31447613460387230382708402325, 7.46689905530641249541725130050, 9.210108221735490576017003424538, 10.48266228613170799006531714575, 10.95675996687122808656202872858, 13.37884870407868643664190259641, 14.01008024990347725690679754779, 14.67334384428117688047836679083