L(s) = 1 | + (0.563 + 0.826i)2-s + (1.61 − 0.624i)3-s + (−0.365 + 0.930i)4-s + (−1.51 − 0.468i)5-s + (1.42 + 0.982i)6-s + (1.45 + 2.20i)7-s + (−0.974 + 0.222i)8-s + (2.21 − 2.01i)9-s + (−0.468 − 1.51i)10-s + (5.27 − 0.395i)11-s + (−0.00854 + 1.73i)12-s + (−1.37 + 2.85i)13-s + (−1.00 + 2.44i)14-s + (−2.74 + 0.191i)15-s + (−0.733 − 0.680i)16-s + (−2.44 − 0.368i)17-s + ⋯ |
L(s) = 1 | + (0.398 + 0.584i)2-s + (0.932 − 0.360i)3-s + (−0.182 + 0.465i)4-s + (−0.678 − 0.209i)5-s + (0.582 + 0.401i)6-s + (0.550 + 0.834i)7-s + (−0.344 + 0.0786i)8-s + (0.739 − 0.672i)9-s + (−0.148 − 0.479i)10-s + (1.59 − 0.119i)11-s + (−0.00246 + 0.499i)12-s + (−0.381 + 0.792i)13-s + (−0.268 + 0.654i)14-s + (−0.708 + 0.0495i)15-s + (−0.183 − 0.170i)16-s + (−0.593 − 0.0894i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87499 + 0.686335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87499 + 0.686335i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.563 - 0.826i)T \) |
| 3 | \( 1 + (-1.61 + 0.624i)T \) |
| 7 | \( 1 + (-1.45 - 2.20i)T \) |
good | 5 | \( 1 + (1.51 + 0.468i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-5.27 + 0.395i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.85i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (2.44 + 0.368i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-3.11 - 1.79i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.00 + 6.64i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (7.22 + 5.76i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (7.87 - 4.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.58 + 4.04i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.05 - 9.00i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.968 + 4.24i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.823 - 0.561i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (2.02 + 0.793i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (8.68 - 2.68i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-6.14 + 2.41i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.545 - 0.434i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-8.39 + 12.3i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.43 + 4.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.44 - 1.65i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.231 - 3.08i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 6.27iT - 97T^{2} \) |
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
−12.04328748552642660087394287197, −11.39451297575372317686318858509, −9.439040719172824853689255904773, −8.900173542101302901227871312520, −8.025344597233113826932566599447, −7.09885533167290913382248783386, −6.10972085998562102440103759590, −4.52224057330415134011031027408, −3.69535389748367293951054983003, −2.01787475705909064711124857913,
1.67719519323134688615337114673, 3.51966836189485098183087479710, 3.95464279701664366315210437397, 5.24157345310777134318451319565, 7.10712947078821316763068978853, 7.74108465748390911923871346648, 9.085637679993651530581880759716, 9.724183006749943479367652597816, 10.98439918809059720156960566608, 11.43012096864663909931180070241