L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.475 + 1.66i)3-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.421 − 1.68i)6-s + (0.0551 + 2.64i)7-s + 0.999i·8-s + (−2.54 − 1.58i)9-s + (−0.866 − 0.499i)10-s + (−0.167 − 0.0969i)11-s + (1.20 + 1.24i)12-s + 1.54i·13-s + (−1.37 − 2.26i)14-s + (−1.68 + 0.421i)15-s + (−0.5 − 0.866i)16-s + (0.264 − 0.458i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.274 + 0.961i)3-s + (0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.171 − 0.685i)6-s + (0.0208 + 0.999i)7-s + 0.353i·8-s + (−0.849 − 0.527i)9-s + (−0.273 − 0.158i)10-s + (−0.0506 − 0.0292i)11-s + (0.347 + 0.359i)12-s + 0.429i·13-s + (−0.366 − 0.604i)14-s + (−0.433 + 0.108i)15-s + (−0.125 − 0.216i)16-s + (0.0642 − 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.235038 + 0.693030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.235038 + 0.693030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.475 - 1.66i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.0551 - 2.64i)T \) |
good | 11 | \( 1 + (0.167 + 0.0969i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.54iT - 13T^{2} \) |
| 17 | \( 1 + (-0.264 + 0.458i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.53 - 3.19i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.87iT - 29T^{2} \) |
| 31 | \( 1 + (-8.02 - 4.63i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.881 + 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + (-4.90 - 8.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0562 - 0.0324i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.01 + 10.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 2.14i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.29iT - 71T^{2} \) |
| 73 | \( 1 + (7.00 + 4.04i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.38 + 5.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + (-8.18 - 14.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.24iT - 97T^{2} \) |
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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
−12.46689044155730558248161609615, −11.54280046298689097064321923056, −10.62945140738245345791326184589, −9.758518362945953989743953402053, −8.963940037010146550500648463777, −7.993947164426558964753131717835, −6.34931859135854209639267159313, −5.71610115050599952354247860053, −4.26770060976838893375887947455, −2.49821820845109492512498500137,
0.791014032461784197469503130700, 2.42933429068951908551094894458, 4.33573602666669652503405282060, 5.98733707327213110240931514355, 7.05362640053560077863239301916, 7.976641021181525209597930649029, 8.871973978818248838976595555346, 10.25877426943876753998061831670, 10.91820128286113159339724185083, 12.02040096907172681690983108686