L(s) = 1 | + 0.715·2-s + (−1.27 + 2.21i)3-s − 1.48·4-s + (0.380 − 0.658i)5-s + (−0.915 + 1.58i)6-s + (−1.25 − 2.16i)7-s − 2.49·8-s + (−1.77 − 3.06i)9-s + (0.272 − 0.471i)10-s + 3.64·11-s + (1.90 − 3.29i)12-s + (−0.946 − 1.63i)13-s + (−0.895 − 1.55i)14-s + (0.971 + 1.68i)15-s + 1.18·16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + 0.506·2-s + (−0.738 + 1.27i)3-s − 0.743·4-s + (0.169 − 0.294i)5-s + (−0.373 + 0.647i)6-s + (−0.473 − 0.819i)7-s − 0.882·8-s + (−0.590 − 1.02i)9-s + (0.0860 − 0.149i)10-s + 1.09·11-s + (0.549 − 0.951i)12-s + (−0.262 − 0.454i)13-s + (−0.239 − 0.414i)14-s + (0.250 + 0.434i)15-s + 0.296·16-s + (0.121 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03602 - 0.177888i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03602 - 0.177888i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-5.24 - 3.94i)T \) |
good | 2 | \( 1 - 0.715T + 2T^{2} \) |
| 3 | \( 1 + (1.27 - 2.21i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.380 + 0.658i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.25 + 2.16i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + (0.946 + 1.63i)T + (-6.5 + 11.2i)T^{2} \) |
| 19 | \( 1 + (-3.30 + 5.72i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.37 - 2.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.32 + 2.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.99 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.162 + 0.280i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + (-4.87 + 8.45i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + (4.32 + 7.49i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.90 + 8.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.74 + 3.01i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.33 - 12.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.69 + 8.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.889 + 1.54i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.92 + 11.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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
−10.21210794905088949558567179622, −9.428299740166385690691100963504, −9.231188529448878037025636362510, −7.74100752759276189601839813131, −6.46156407031159995931443052513, −5.57902917481545513561798381948, −4.76864946720457347648302997217, −4.07893297390514749577168357597, −3.29603508796149165184346272788, −0.61757441131443800109988068587,
1.21401756038726033894393629545, 2.72764148033239938425832327546, 4.02443960501742115813445897447, 5.29097881563285495549911566949, 6.10481736254822238027626752557, 6.57458921213352266680712142762, 7.65867728508925520402844189788, 8.777273791223622777636664345104, 9.451153807291200739354599395026, 10.50801227986753015120016066454