L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.15 − 1.29i)3-s + (0.499 + 0.866i)4-s + 5-s + (0.353 + 1.69i)6-s + (2.60 − 0.461i)7-s − 0.999i·8-s + (−0.335 + 2.98i)9-s + (−0.866 − 0.5i)10-s + 5.17i·11-s + (0.541 − 1.64i)12-s + (0.604 + 0.349i)13-s + (−2.48 − 0.903i)14-s + (−1.15 − 1.29i)15-s + (−0.5 + 0.866i)16-s + (−1.77 + 3.07i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.666 − 0.745i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.144 + 0.692i)6-s + (0.984 − 0.174i)7-s − 0.353i·8-s + (−0.111 + 0.993i)9-s + (−0.273 − 0.158i)10-s + 1.56i·11-s + (0.156 − 0.474i)12-s + (0.167 + 0.0968i)13-s + (−0.664 − 0.241i)14-s + (−0.298 − 0.333i)15-s + (−0.125 + 0.216i)16-s + (−0.430 + 0.745i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.897136 + 0.165245i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.897136 + 0.165245i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (1.15 + 1.29i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.60 + 0.461i)T \) |
good | 11 | \( 1 - 5.17iT - 11T^{2} \) |
| 13 | \( 1 + (-0.604 - 0.349i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.77 - 3.07i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.75iT - 23T^{2} \) |
| 29 | \( 1 + (4.60 - 2.65i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.81 + 2.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.06 - 8.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.18 - 5.52i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.94 + 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.08 + 7.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.82 - 5.09i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.679 + 1.17i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.28 - 3.05i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.21 - 3.82i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (3.19 + 1.84i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.81 + 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.80 + 13.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.57 - 4.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.5 + 9.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.50344298579275996469281626560, −10.11181726155541137946939942749, −8.824010729128065838959496061063, −7.962756098408274042975655499072, −7.21391748961962009909966444797, −6.33935938341543702291014246336, −5.19259994235605263013334030897, −4.17096593408388289705328870489, −2.12173334113316940270540412824, −1.54477453360663689123926194839,
0.69317309493941152292689455596, 2.56050516586769730899173571619, 4.21441447388559845984657548087, 5.22342658825373627022978796436, 5.99582266062224355345388912713, 6.80409784534026841011606275918, 8.247947490864548372964339650372, 8.779344709681736467647277654404, 9.613198939599907183094099333963, 10.79366588350160428283741453126