L(s) = 1 | + (−2.47 + 0.664i)2-s + (−2.33 − 1.34i)3-s + (3.97 − 2.29i)4-s + (−0.281 + 0.281i)5-s + (6.68 + 1.79i)6-s + (−0.201 + 2.63i)7-s + (−4.70 + 4.70i)8-s + (2.13 + 3.69i)9-s + (0.511 − 0.885i)10-s + (0.939 + 3.50i)11-s − 12.3·12-s + (3.44 − 1.06i)13-s + (−1.25 − 6.67i)14-s + (1.03 − 0.277i)15-s + (3.95 − 6.84i)16-s + (2.04 + 3.54i)17-s + ⋯ |
L(s) = 1 | + (−1.75 + 0.469i)2-s + (−1.34 − 0.778i)3-s + (1.98 − 1.14i)4-s + (−0.125 + 0.125i)5-s + (2.72 + 0.731i)6-s + (−0.0762 + 0.997i)7-s + (−1.66 + 1.66i)8-s + (0.711 + 1.23i)9-s + (0.161 − 0.280i)10-s + (0.283 + 1.05i)11-s − 3.57·12-s + (0.955 − 0.294i)13-s + (−0.334 − 1.78i)14-s + (0.267 − 0.0717i)15-s + (0.987 − 1.71i)16-s + (0.496 + 0.860i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.211409 + 0.159968i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.211409 + 0.159968i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.201 - 2.63i)T \) |
| 13 | \( 1 + (-3.44 + 1.06i)T \) |
good | 2 | \( 1 + (2.47 - 0.664i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (2.33 + 1.34i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.281 - 0.281i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.939 - 3.50i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.04 - 3.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.777 - 0.208i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (4.41 + 2.54i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.44 - 4.44i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.463 - 1.73i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.578 - 2.15i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.99 - 5.99i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.09T + 53T^{2} \) |
| 59 | \( 1 + (-1.92 + 7.17i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.40 + 1.38i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.85 - 1.30i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.582 - 2.17i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (3.50 + 3.50i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + (-5.36 + 5.36i)T - 83iT^{2} \) |
| 89 | \( 1 + (11.5 - 3.09i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (7.71 + 2.06i)T + (84.0 + 48.5i)T^{2} \) |
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
−14.85272682526620287325950075536, −12.73483277039187037011100775053, −11.87072520786540614395082968481, −10.96508151217778519651397616163, −9.954554373756480914244718751870, −8.661948440820426965308153512231, −7.52928619440171145337588151698, −6.46254752818505259102035904807, −5.67282568787218822129103138379, −1.61441421437996250762728169136,
0.70185231427497952996475074318, 3.83105175680969102623252915648, 5.92264203592299581365821186747, 7.22454727977435344152395567589, 8.606071694182691849422884525544, 9.803238659822800104817668154080, 10.57298152495110894818964834640, 11.32437363717416910052781257599, 11.96015417482872792041805758248, 13.77907857433001419092204033539