L(s) = 1 | + (4.91 + 2.83i)5-s + (−7.94 − 13.7i)11-s + 20.1i·13-s + (0.823 − 0.475i)17-s + (−27.5 − 15.9i)19-s + (13 − 22.5i)23-s + (3.60 + 6.23i)25-s − 27.7·29-s + (−13.1 + 7.57i)31-s + (16 − 27.7i)37-s − 17.3i·41-s − 59.2·43-s + (−66.1 − 38.1i)47-s + (12.8 + 22.3i)53-s − 90.2i·55-s + ⋯ |
L(s) = 1 | + (0.982 + 0.567i)5-s + (−0.722 − 1.25i)11-s + 1.55i·13-s + (0.0484 − 0.0279i)17-s + (−1.45 − 0.838i)19-s + (0.565 − 0.978i)23-s + (0.144 + 0.249i)25-s − 0.958·29-s + (−0.423 + 0.244i)31-s + (0.432 − 0.748i)37-s − 0.422i·41-s − 1.37·43-s + (−1.40 − 0.812i)47-s + (0.243 + 0.421i)53-s − 1.64i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8357363738\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8357363738\) |
\(L(2)\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (-4.91 - 2.83i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (7.94 + 13.7i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 20.1iT - 169T^{2} \) |
| 17 | \( 1 + (-0.823 + 0.475i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (27.5 + 15.9i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-13 + 22.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 27.7T + 841T^{2} \) |
| 31 | \( 1 + (13.1 - 7.57i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-16 + 27.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 17.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (66.1 + 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-12.8 - 22.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-59.2 + 34.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-46.9 - 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (43.6 + 75.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-60.9 + 35.1i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (20.1 - 34.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 71.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (66.9 + 38.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 128. iT - 9.40e3T^{2} \) |
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
−8.860594984619361917854777209202, −8.248542584954082176956312687313, −6.92252347110782626203427819654, −6.54194435217873537202607022051, −5.70241464457058223504633160280, −4.79898758667429711355187445539, −3.71981932731047642782672522503, −2.57962012607466835542330736112, −1.90119443438419738761162314692, −0.19721053128674291811314945057,
1.43281640718091463804636531235, 2.26659924088150940239175126674, 3.42774675229964699049481313543, 4.66302027174591465749856452071, 5.38100253574697472617338035795, 5.96183897223752489717683843462, 7.05486181165001048653453339413, 7.915814707408871952144781626338, 8.521054795595783690262471689110, 9.673610699828253289983264393486