| L(s) = 1 | − 3.46·5-s + (−1 − 2.44i)7-s − 4.24i·11-s + 4.89i·13-s + 3.46·17-s − 4.89i·19-s − 4.24i·23-s + 6.99·25-s − 4.24i·29-s + (3.46 + 8.48i)35-s + 8·37-s − 3.46·41-s + 2·43-s − 6.92·47-s + (−4.99 + 4.89i)49-s + ⋯ |
| L(s) = 1 | − 1.54·5-s + (−0.377 − 0.925i)7-s − 1.27i·11-s + 1.35i·13-s + 0.840·17-s − 1.12i·19-s − 0.884i·23-s + 1.39·25-s − 0.787i·29-s + (0.585 + 1.43i)35-s + 1.31·37-s − 0.541·41-s + 0.304·43-s − 1.01·47-s + (−0.714 + 0.699i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4059086643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4059086643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 4.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
| 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
−8.166739465355602272527361030698, −7.21967933738747994869140979771, −6.82811756295447267142268729469, −5.96540234377211939937171421846, −4.74753056007180436897972517996, −4.13612180376950093685972136455, −3.55293469435628186039436806496, −2.69692223347604779980805550460, −0.996914947118759936075404499524, −0.14913286812657322377418442719,
1.37088800537173436441269826180, 2.75331133160320226861115906339, 3.42996128020740950563017540543, 4.17876653985352456096687513964, 5.16347309132208418476105271848, 5.73780135450459054595467913766, 6.77518266228749379400094940219, 7.70420779364822210818197621047, 7.84689305409143072616658253365, 8.659648130811361421003094994546
