L(s) = 1 | + (−1.93 − 1.19i)2-s + (−0.263 + 1.41i)3-s + (1.41 + 2.83i)4-s + (3.28 − 0.304i)5-s + (2.19 − 2.41i)6-s + (0.242 − 0.625i)7-s + (0.245 − 2.65i)8-s + (0.876 + 0.339i)9-s + (−6.71 − 3.34i)10-s + (−3.96 + 1.97i)11-s + (−4.37 + 1.24i)12-s + (−0.0517 + 0.558i)13-s + (−1.21 + 0.919i)14-s + (−0.436 + 4.71i)15-s + (0.171 − 0.227i)16-s + (1.05 + 3.98i)17-s + ⋯ |
L(s) = 1 | + (−1.36 − 0.846i)2-s + (−0.152 + 0.814i)3-s + (0.706 + 1.41i)4-s + (1.46 − 0.136i)5-s + (0.897 − 0.984i)6-s + (0.0916 − 0.236i)7-s + (0.0869 − 0.937i)8-s + (0.292 + 0.113i)9-s + (−2.12 − 1.05i)10-s + (−1.19 + 0.595i)11-s + (−1.26 + 0.359i)12-s + (−0.0143 + 0.154i)13-s + (−0.325 + 0.245i)14-s + (−0.112 + 1.21i)15-s + (0.0429 − 0.0568i)16-s + (0.256 + 0.966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.789213 + 0.0889459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.789213 + 0.0889459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-1.05 - 3.98i)T \) |
good | 2 | \( 1 + (1.93 + 1.19i)T + (0.891 + 1.79i)T^{2} \) |
| 3 | \( 1 + (0.263 - 1.41i)T + (-2.79 - 1.08i)T^{2} \) |
| 5 | \( 1 + (-3.28 + 0.304i)T + (4.91 - 0.918i)T^{2} \) |
| 7 | \( 1 + (-0.242 + 0.625i)T + (-5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (3.96 - 1.97i)T + (6.62 - 8.77i)T^{2} \) |
| 13 | \( 1 + (0.0517 - 0.558i)T + (-12.7 - 2.38i)T^{2} \) |
| 19 | \( 1 + (-2.85 + 1.76i)T + (8.46 - 17.0i)T^{2} \) |
| 23 | \( 1 + (-0.0390 + 0.100i)T + (-16.9 - 15.4i)T^{2} \) |
| 29 | \( 1 + (-2.07 + 1.03i)T + (17.4 - 23.1i)T^{2} \) |
| 31 | \( 1 + (-4.72 - 0.438i)T + (30.4 + 5.69i)T^{2} \) |
| 37 | \( 1 + (-5.29 - 1.50i)T + (31.4 + 19.4i)T^{2} \) |
| 41 | \( 1 + (-0.482 + 2.57i)T + (-38.2 - 14.8i)T^{2} \) |
| 43 | \( 1 + (-2.03 - 2.69i)T + (-11.7 + 41.3i)T^{2} \) |
| 47 | \( 1 + (6.92 - 2.68i)T + (34.7 - 31.6i)T^{2} \) |
| 53 | \( 1 + (-0.673 - 0.260i)T + (39.1 + 35.7i)T^{2} \) |
| 59 | \( 1 + (-8.49 + 7.74i)T + (5.44 - 58.7i)T^{2} \) |
| 61 | \( 1 + (-0.416 + 0.456i)T + (-5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 0.943i)T + (29.8 - 59.9i)T^{2} \) |
| 71 | \( 1 + (3.78 - 9.76i)T + (-52.4 - 47.8i)T^{2} \) |
| 73 | \( 1 + (13.0 + 9.86i)T + (19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (6.49 + 10.4i)T + (-35.2 + 70.7i)T^{2} \) |
| 83 | \( 1 + (17.2 - 3.22i)T + (77.3 - 29.9i)T^{2} \) |
| 89 | \( 1 + (0.596 + 6.43i)T + (-87.4 + 16.3i)T^{2} \) |
| 97 | \( 1 + (2.14 - 5.53i)T + (-71.6 - 65.3i)T^{2} \) |
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
−11.32689459728340229581754526384, −10.40447175173333838255798746563, −10.02861532446614682766442576240, −9.494854049237369938705054637296, −8.393330199681178436888307397291, −7.31625089119502875736557242237, −5.74806199462459265143090840988, −4.63558904066712387975700061705, −2.77197558820454862689836271206, −1.57692747082740778426546747439,
1.07012850689366620700662957768, 2.52127657400966765848774565017, 5.40101858802446945367963590912, 6.08078817727957569563691172149, 7.04493149144012854720736139260, 7.84364560736078501148733083075, 8.827989482323919919310544484222, 9.920796139400214968438598633338, 10.20668140871157807685071095550, 11.59955699910699189515827945637