L(s) = 1 | + (−1.32 + 1.11i)3-s + (1 + 1.73i)4-s + (1.93 − 1.11i)5-s + (1.32 − 2.29i)7-s + (0.5 − 2.95i)9-s + (4.81 + 2.77i)11-s + (−3.25 − 1.17i)12-s + (−2.24 − 3.88i)13-s + (−1.31 + 3.64i)15-s + (−1.99 + 3.46i)16-s + 7.99i·17-s + (3.87 + 2.23i)20-s + (0.811 + 4.51i)21-s + (2.5 − 4.33i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯ |
L(s) = 1 | + (−0.763 + 0.645i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.499i)5-s + (0.499 − 0.866i)7-s + (0.166 − 0.986i)9-s + (1.45 + 0.837i)11-s + (−0.940 − 0.338i)12-s + (−0.622 − 1.07i)13-s + (−0.338 + 0.940i)15-s + (−0.499 + 0.866i)16-s + 1.93i·17-s + (0.866 + 0.499i)20-s + (0.177 + 0.984i)21-s + (0.5 − 0.866i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33039 + 0.489568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33039 + 0.489568i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.32 - 1.11i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (-1.32 + 2.29i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-4.81 - 2.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.24 + 3.88i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7.99iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.12 + 2.95i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.11 + 0.641i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.47iT - 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.03 - 4.63i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-1.72 + 2.98i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.82086880813512354525343044294, −10.74268590359134221964454412764, −10.09901109488566610696867856050, −9.080771884810528296551160990715, −7.920372315500895212465058678690, −6.78821212216472374206624964730, −5.88885115331051703871566596164, −4.54638053923673442310430910499, −3.73659200235889966652948328189, −1.65324906473425603656228003963,
1.44959440737507015734992759619, 2.51679075134356188061877534772, 4.95018179885979440676735078639, 5.75418866242043665267494464575, 6.59202225214968113971492661388, 7.22328036090298838829998491748, 9.058469786663986906107715853747, 9.630020995799916829820518789037, 10.96465101405556993512019567983, 11.54337807319509455906435694940