| L(s) = 1 | + (−1.00 + 0.177i)2-s + (2.83 − 0.987i)3-s + (−2.77 + 1.01i)4-s + (1.20 + 1.44i)5-s + (−2.67 + 1.49i)6-s + (0.994 + 1.72i)7-s + (6.15 − 3.55i)8-s + (7.04 − 5.59i)9-s + (−1.47 − 1.23i)10-s + 9.99i·11-s + (−6.87 + 5.60i)12-s + (13.9 + 11.6i)13-s + (−1.30 − 1.55i)14-s + (4.85 + 2.88i)15-s + (3.49 − 2.93i)16-s + (2.78 + 3.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.503 + 0.0887i)2-s + (0.944 − 0.329i)3-s + (−0.694 + 0.252i)4-s + (0.241 + 0.288i)5-s + (−0.445 + 0.249i)6-s + (0.142 + 0.245i)7-s + (0.769 − 0.444i)8-s + (0.783 − 0.621i)9-s + (−0.147 − 0.123i)10-s + 0.908i·11-s + (−0.572 + 0.467i)12-s + (1.06 + 0.897i)13-s + (−0.0932 − 0.111i)14-s + (0.323 + 0.192i)15-s + (0.218 − 0.183i)16-s + (0.164 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42438 + 0.369954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42438 + 0.369954i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.83 + 0.987i)T \) |
| 19 | \( 1 + (-2.79 + 18.7i)T \) |
| good | 2 | \( 1 + (1.00 - 0.177i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.20 - 1.44i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-0.994 - 1.72i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 9.99iT - 121T^{2} \) |
| 13 | \( 1 + (-13.9 - 11.6i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-2.78 - 3.32i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-11.7 - 32.2i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-8.41 - 23.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 13.8T + 961T^{2} \) |
| 37 | \( 1 - 35.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (20.3 - 3.59i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (51.3 + 18.6i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (8.64 + 23.7i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (81.3 + 14.3i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-5.37 + 14.7i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (54.9 + 46.0i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (10.0 - 57.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (66.1 - 11.6i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (78.3 + 28.5i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-72.9 + 61.1i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-34.3 + 19.8i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (28.4 + 78.1i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-19.1 - 108. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
| 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
−12.92507271383486302145570204134, −11.67660776559363286625761119589, −10.23682700156384972009660292164, −9.287212241594217294449633935670, −8.701319315040240231902710205556, −7.58719671984699926792557405610, −6.65710244315633271986532760990, −4.72492536198893847558152806673, −3.42997460370496397862156764124, −1.64378495508944650344646665092,
1.21078344444754461230279622847, 3.27740769672614254213476886965, 4.58179774221903903195913342963, 5.89264841222910676406666017167, 7.83765387192278311219199050213, 8.457579928247432414900466427044, 9.289930751985899347142816008880, 10.28641946104554390497585549132, 10.99863940267730587801736100060, 12.83258540902932749449124060736
