L(s) = 1 | + (−0.134 + 0.232i)2-s + (0.964 + 1.66i)4-s + (−1.28 − 2.21i)5-s + (0.773 − 2.53i)7-s − 1.05·8-s + 0.686·10-s − 3.94·11-s + (−3.15 − 1.74i)13-s + (0.483 + 0.518i)14-s + (−1.78 + 3.09i)16-s + (0.392 + 0.679i)17-s − 7.49·19-s + (2.46 − 4.27i)20-s + (0.529 − 0.916i)22-s + (−3.97 + 6.88i)23-s + ⋯ |
L(s) = 1 | + (−0.0947 + 0.164i)2-s + (0.482 + 0.834i)4-s + (−0.572 − 0.992i)5-s + (0.292 − 0.956i)7-s − 0.372·8-s + 0.217·10-s − 1.18·11-s + (−0.874 − 0.484i)13-s + (0.129 + 0.138i)14-s + (−0.446 + 0.773i)16-s + (0.0952 + 0.164i)17-s − 1.71·19-s + (0.552 − 0.956i)20-s + (0.112 − 0.195i)22-s + (−0.829 + 1.43i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0628221 - 0.274984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0628221 - 0.274984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.773 + 2.53i)T \) |
| 13 | \( 1 + (3.15 + 1.74i)T \) |
good | 2 | \( 1 + (0.134 - 0.232i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 17 | \( 1 + (-0.392 - 0.679i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 + (3.97 - 6.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.17 - 2.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 2.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.21 + 2.11i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.63 + 8.03i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.48 + 7.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 + (-6.15 + 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.384 - 0.665i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 + (-3.83 + 6.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 2.05i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.992243941499556217655029809543, −8.672024602203511418191029707901, −8.002406836039047699947064074479, −7.62690596135082327611186767297, −6.60672556392443303164719119000, −5.25647587970643413979742658290, −4.39545769916176990706113522073, −3.46956328253061759701443117298, −2.07716585035774655158532891346, −0.12724036247836053559983253282,
2.29194886815494696517848460912, 2.62658950299594464058733565969, 4.35781079850898790315810194801, 5.38509929284682283016831249719, 6.31786938083971465354339408729, 7.04838176937198225977675783637, 8.042683891231017875904552872253, 8.910891919773933205319540817921, 10.07938523303857443975005162382, 10.58035716769575771039378781472