L(s) = 1 | + (−0.440 + 0.863i)2-s + (−1.57 + 0.713i)3-s + (0.622 + 0.857i)4-s + (0.0788 − 1.67i)6-s + (−0.0473 + 0.298i)7-s + (−2.93 + 0.464i)8-s + (1.98 − 2.25i)9-s + (0.774 − 3.22i)11-s + (−1.59 − 0.909i)12-s + (2.68 − 5.27i)13-s + (−0.237 − 0.172i)14-s + (0.234 − 0.720i)16-s + (3.07 − 1.56i)17-s + (1.07 + 2.70i)18-s + (−0.811 + 1.11i)19-s + ⋯ |
L(s) = 1 | + (−0.311 + 0.610i)2-s + (−0.911 + 0.411i)3-s + (0.311 + 0.428i)4-s + (0.0321 − 0.684i)6-s + (−0.0178 + 0.112i)7-s + (−1.03 + 0.164i)8-s + (0.661 − 0.750i)9-s + (0.233 − 0.972i)11-s + (−0.460 − 0.262i)12-s + (0.746 − 1.46i)13-s + (−0.0634 − 0.0460i)14-s + (0.0585 − 0.180i)16-s + (0.746 − 0.380i)17-s + (0.252 + 0.637i)18-s + (−0.186 + 0.256i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.974652 + 0.188587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.974652 + 0.188587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.57 - 0.713i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-0.774 + 3.22i)T \) |
good | 2 | \( 1 + (0.440 - 0.863i)T + (-1.17 - 1.61i)T^{2} \) |
| 7 | \( 1 + (0.0473 - 0.298i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.68 + 5.27i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.07 + 1.56i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (0.811 - 1.11i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.40 + 4.40i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.39 - 1.01i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.40 + 7.39i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.0288 + 0.182i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.87 - 3.96i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (1.62 + 1.62i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.484 - 3.05i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-2.91 - 1.48i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.68 + 1.94i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.59 + 11.0i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (9.94 - 9.94i)T - 67iT^{2} \) |
| 71 | \( 1 + (-7.73 - 2.51i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.1 - 1.60i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (8.06 - 2.62i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.89 + 3.72i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + (-4.32 - 2.20i)T + (57.0 + 78.4i)T^{2} \) |
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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
−10.39585169353904872338633341835, −9.332164923965160545826431202897, −8.476717800041687629664113817753, −7.72646874542898638089766952393, −6.69829509543477162987315366397, −5.91288523223184616678760976031, −5.35724330622986319938930687386, −3.83002138520634294899711054216, −2.97111098468450623919530818812, −0.70091885571792838813186650943,
1.24822383407384700637372507163, 2.01620755620204663535463595459, 3.69205336902623071511051366043, 4.92569456852072804822243336026, 5.85419541647057307846704712178, 6.79553618824101630565909888107, 7.24075364562215131380365603546, 8.749251976823748541621574849280, 9.539487271953434865656081800461, 10.37575220361995064944735939089