| L(s) = 1 | + (−3.18 + 0.676i)3-s + (−0.412 + 2.60i)5-s + (−3.92 + 2.54i)7-s + (6.92 − 3.08i)9-s + (−2.46 + 2.22i)11-s + (−0.584 + 3.55i)13-s + (−0.448 − 8.55i)15-s + (0.189 + 1.80i)17-s + (0.154 − 2.95i)19-s + (10.7 − 10.7i)21-s + (1.36 − 0.786i)23-s + (−1.84 − 0.598i)25-s + (−12.0 + 8.74i)27-s + (−4.29 − 3.87i)29-s + (7.06 − 1.11i)31-s + ⋯ |
| L(s) = 1 | + (−1.83 + 0.390i)3-s + (−0.184 + 1.16i)5-s + (−1.48 + 0.963i)7-s + (2.30 − 1.02i)9-s + (−0.742 + 0.669i)11-s + (−0.162 + 0.986i)13-s + (−0.115 − 2.20i)15-s + (0.0459 + 0.436i)17-s + (0.0355 − 0.677i)19-s + (2.34 − 2.34i)21-s + (0.284 − 0.164i)23-s + (−0.368 − 0.119i)25-s + (−2.31 + 1.68i)27-s + (−0.798 − 0.718i)29-s + (1.26 − 0.200i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0908484 - 0.117839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0908484 - 0.117839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (2.46 - 2.22i)T \) |
| 13 | \( 1 + (0.584 - 3.55i)T \) |
| good | 3 | \( 1 + (3.18 - 0.676i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (0.412 - 2.60i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (3.92 - 2.54i)T + (2.84 - 6.39i)T^{2} \) |
| 17 | \( 1 + (-0.189 - 1.80i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.154 + 2.95i)T + (-18.8 - 1.98i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 0.786i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.29 + 3.87i)T + (3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (-7.06 + 1.11i)T + (29.4 - 9.57i)T^{2} \) |
| 37 | \( 1 + (7.37 - 0.386i)T + (36.7 - 3.86i)T^{2} \) |
| 41 | \( 1 + (1.28 + 0.832i)T + (16.6 + 37.4i)T^{2} \) |
| 43 | \( 1 + (-2.46 + 4.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.373i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.294 + 0.214i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-8.30 - 12.7i)T + (-23.9 + 53.8i)T^{2} \) |
| 61 | \( 1 + (8.12 - 0.854i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.65 - 0.710i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.48 + 1.19i)T + (14.7 + 69.4i)T^{2} \) |
| 73 | \( 1 + (-5.78 - 2.94i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (3.48 - 4.79i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.1 - 1.60i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.136 + 0.508i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.65 + 12.1i)T + (-72.0 - 64.9i)T^{2} \) |
| 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
−11.36385237497277398623577493666, −10.44161260267627384556885603597, −9.985555889289784764860583701267, −9.105271661026290979766046188887, −7.22503168581169053683189198622, −6.66095073861887009720364852487, −6.02995460945107032306403699102, −5.07254980881859498821525804049, −3.87882108496678904026559685299, −2.51770223793833026196395598629,
0.13554695920106750448745292543, 0.982611716085993855086470454125, 3.44636695974824593071241086388, 4.78812817178051855088060397420, 5.47664260290006389387994477855, 6.32979239486696183358605088209, 7.20942320189598865820033075057, 8.116861634959211416785011210230, 9.559225557736422540072537688604, 10.34033043654460899998431939282
