| L(s) = 1 | + (−1.39 + 0.247i)2-s + (1.73 + 0.0113i)3-s + (1.87 − 0.689i)4-s + (3.55 + 0.626i)5-s + (−2.41 + 0.413i)6-s + (0.197 − 0.165i)7-s + (−2.44 + 1.42i)8-s + (2.99 + 0.0393i)9-s + (−5.10 + 0.00789i)10-s + (−4.72 + 0.833i)11-s + (3.25 − 1.17i)12-s + (−2.06 − 5.67i)13-s + (−0.233 + 0.279i)14-s + (6.15 + 1.12i)15-s + (3.04 − 2.59i)16-s + (−2.76 + 4.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.984 + 0.175i)2-s + (0.999 + 0.00655i)3-s + (0.938 − 0.344i)4-s + (1.58 + 0.280i)5-s + (−0.985 + 0.168i)6-s + (0.0746 − 0.0626i)7-s + (−0.863 + 0.504i)8-s + (0.999 + 0.0131i)9-s + (−1.61 + 0.00249i)10-s + (−1.42 + 0.251i)11-s + (0.940 − 0.338i)12-s + (−0.573 − 1.57i)13-s + (−0.0625 + 0.0747i)14-s + (1.58 + 0.290i)15-s + (0.762 − 0.647i)16-s + (−0.670 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28112 + 0.139995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28112 + 0.139995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.247i)T \) |
| 3 | \( 1 + (-1.73 - 0.0113i)T \) |
| good | 5 | \( 1 + (-3.55 - 0.626i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.197 + 0.165i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (4.72 - 0.833i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.06 + 5.67i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.75 - 1.01i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.164 + 0.137i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.455 + 1.25i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.00197 + 0.00165i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.995 + 0.574i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.72 + 1.72i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.99 - 0.881i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.18 - 0.993i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 + (10.3 + 1.81i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.31 - 6.33i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.10 + 3.02i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.40 - 5.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.51 - 2.61i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 3.69i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.84 + 7.80i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.381 + 0.660i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.902 + 5.11i)T + (-91.1 + 33.1i)T^{2} \) |
| 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
−12.74212245407199326076483736661, −10.57140332067653880941297056362, −10.34368535170765502515189665659, −9.504042031386394717696507935795, −8.373068268114727246697181433218, −7.65570437408752901585340285325, −6.37637479395557236576601112753, −5.28902632732136786610178865109, −2.84077932279716046415521555338, −1.98595771513531620479358293654,
1.94472213997760597804707226960, 2.68954075545570027790532684328, 4.88517760740943221223306035218, 6.45337414170214537699630672223, 7.43038979659609279780761447578, 8.681856538466882512184533730999, 9.333451199594679623175759320731, 9.962772759874453835419016910680, 10.95886182521715580292208142666, 12.35171511542067439273027179389
