L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 2.70·7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (−4.54 − 3.30i)11-s + (−0.809 + 0.587i)12-s + (3.91 − 2.84i)13-s + (−2.19 − 1.59i)14-s + (−0.809 + 0.587i)16-s + (0.323 − 0.994i)17-s + 0.999·18-s + (2.59 − 7.97i)19-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.126 − 0.388i)6-s + 1.02·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (−1.37 − 0.996i)11-s + (−0.233 + 0.169i)12-s + (1.08 − 0.789i)13-s + (−0.585 − 0.425i)14-s + (−0.202 + 0.146i)16-s + (0.0783 − 0.241i)17-s + 0.235·18-s + (0.594 − 1.82i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24414 - 0.408037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24414 - 0.408037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 + (4.54 + 3.30i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.91 + 2.84i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.323 + 0.994i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 7.97i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.28 - 3.11i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.29 - 3.98i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.72 - 5.30i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.42 + 1.75i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.27 + 0.927i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + (0.949 + 2.92i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.90 - 5.87i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-11.4 + 8.33i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.218 + 0.159i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.94 + 6.00i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.40 + 4.31i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.60 - 4.79i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.85 - 11.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.0415 - 0.127i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (9.53 + 6.93i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.815 + 2.51i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.49537964333023968603476210321, −9.328459711769699622886859492496, −8.581802317336256681571001904842, −8.027938681139898383322900834850, −7.06670480291286970253805156556, −5.49374286170138324845397446600, −4.95064547303919728788383499263, −3.43808676188391083283692866043, −2.67005433781728781890117692045, −0.921411858983269402580375462134,
1.36162412449107342300609611329, 2.37280485074117234746475233071, 4.11096783255142614562606988545, 5.25542814099191686310335651895, 6.13290508736531991398295427599, 7.22032498210546051850863773408, 7.973575670136541701731159668307, 8.373467609090364285691269819731, 9.528211857868232728008411080138, 10.36516892941212016646746732136