L(s) = 1 | + (−0.258 + 0.965i)2-s − i·3-s + (−0.866 − 0.499i)4-s + (0.782 + 2.92i)5-s + (0.965 + 0.258i)6-s + (1.58 + 2.12i)7-s + (0.707 − 0.707i)8-s − 9-s − 3.02·10-s + (−1.55 + 1.55i)11-s + (−0.499 + 0.866i)12-s + (1.15 − 3.41i)13-s + (−2.45 + 0.977i)14-s + (2.92 − 0.782i)15-s + (0.500 + 0.866i)16-s + (−2.40 + 4.15i)17-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s − 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.349 + 1.30i)5-s + (0.394 + 0.105i)6-s + (0.597 + 0.801i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 0.956·10-s + (−0.469 + 0.469i)11-s + (−0.144 + 0.249i)12-s + (0.319 − 0.947i)13-s + (−0.657 + 0.261i)14-s + (0.754 − 0.202i)15-s + (0.125 + 0.216i)16-s + (−0.582 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.636347 + 1.04349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.636347 + 1.04349i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-1.58 - 2.12i)T \) |
| 13 | \( 1 + (-1.15 + 3.41i)T \) |
good | 5 | \( 1 + (-0.782 - 2.92i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.55 - 1.55i)T - 11iT^{2} \) |
| 17 | \( 1 + (2.40 - 4.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.01 - 2.01i)T - 19iT^{2} \) |
| 23 | \( 1 + (0.721 - 0.416i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.310 + 0.538i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.04 - 0.547i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.98 - 1.06i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.82 - 10.5i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (5.69 - 3.28i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.82 + 1.02i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.61 - 9.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.61 - 2.03i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 7.60iT - 61T^{2} \) |
| 67 | \( 1 + (5.74 + 5.74i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.25 + 8.42i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.17 + 15.5i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.859 + 1.48i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.5 + 12.5i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.91 + 7.15i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-11.7 - 3.14i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.81604557682574242933308909077, −10.37602152247679049645717416049, −9.168370509441716658668943711326, −8.072838555003358344681967842143, −7.66166540444287374247153563227, −6.28176291356835042031083959578, −6.10103913453137689104305803863, −4.74811448818012305456438148645, −3.06203322602245458375744916230, −1.92086041713149482841574604489,
0.76938299441377568005771217321, 2.25290632399791107820493685633, 3.94609184934572112559005793019, 4.65650155163438766049245206565, 5.45063938277547667545134661714, 6.99747570743713902619402938087, 8.293858571979558336784887099926, 8.876173780323549236977846575852, 9.598177902649658583924548500735, 10.57353122377061627352461408388