L(s) = 1 | + (−2.00 − 1.49i)2-s + (−1.71 − 0.269i)3-s + (1.22 + 4.07i)4-s + (−3.15 − 2.07i)5-s + (3.03 + 3.09i)6-s + (−0.572 + 2.58i)7-s + (1.92 − 5.29i)8-s + (2.85 + 0.921i)9-s + (3.22 + 8.86i)10-s + (−0.664 + 1.32i)11-s + (−0.990 − 7.30i)12-s + (−2.20 − 0.950i)13-s + (5.00 − 4.32i)14-s + (4.83 + 4.39i)15-s + (−4.67 + 3.07i)16-s + (4.99 + 4.19i)17-s + ⋯ |
L(s) = 1 | + (−1.41 − 1.05i)2-s + (−0.987 − 0.155i)3-s + (0.610 + 2.03i)4-s + (−1.40 − 0.927i)5-s + (1.23 + 1.26i)6-s + (−0.216 + 0.976i)7-s + (0.681 − 1.87i)8-s + (0.951 + 0.307i)9-s + (1.02 + 2.80i)10-s + (−0.200 + 0.399i)11-s + (−0.285 − 2.10i)12-s + (−0.610 − 0.263i)13-s + (1.33 − 1.15i)14-s + (1.24 + 1.13i)15-s + (−1.16 + 0.768i)16-s + (1.21 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0903535 - 0.195958i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0903535 - 0.195958i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 + 0.269i)T \) |
| 7 | \( 1 + (0.572 - 2.58i)T \) |
good | 2 | \( 1 + (2.00 + 1.49i)T + (0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (3.15 + 2.07i)T + (1.98 + 4.59i)T^{2} \) |
| 11 | \( 1 + (0.664 - 1.32i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (2.20 + 0.950i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (-4.99 - 4.19i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 1.41i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (4.02 + 3.79i)T + (1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.617 - 5.28i)T + (-28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-1.44 + 6.08i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (0.361 - 2.05i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-2.35 - 3.15i)T + (-11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (0.560 + 9.62i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (-0.725 + 0.171i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (9.25 + 5.34i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.37 - 3.20i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-8.54 - 2.55i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (8.70 - 1.01i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (1.67 + 4.61i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.43 - 3.95i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.48 + 10.0i)T + (-22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-3.78 + 5.08i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (2.75 + 1.00i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.709 + 1.07i)T + (-38.4 + 89.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37641617309941259409949110358, −9.757356024856667061751578512831, −8.674916948875938724099731544381, −7.989777826391946412681728122637, −7.37036277900977322909118080166, −5.80943294595717950771454357747, −4.61693634219953470792233441203, −3.40624765938746000983560949208, −1.77821073877370288266200725328, −0.36129652779759643369666350685,
0.77480310919779160854727642167, 3.44808156016529854045197459501, 4.76122847096017061207225702004, 6.03291809525129574388038878712, 6.91977969141122098659187773649, 7.53172632384025962005645792520, 7.896732615493299272396925505131, 9.504523002496719091501465634203, 10.07046296370333903806598606386, 10.88721809654059647995413562106