| L(s) = 1 | + (0.654 − 0.755i)3-s + (2.26 − 1.45i)5-s + (−0.275 − 0.110i)7-s + (−0.142 − 0.989i)9-s + (−0.141 − 2.97i)11-s + (−2.35 + 0.225i)13-s + (0.383 − 2.66i)15-s + (−1.55 − 6.43i)17-s + (−0.0392 + 0.0157i)19-s + (−0.264 + 0.136i)21-s + (6.70 + 1.29i)23-s + (0.932 − 2.04i)25-s + (−0.841 − 0.540i)27-s + (−4.54 + 7.87i)29-s + (6.70 + 0.640i)31-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.436i)3-s + (1.01 − 0.650i)5-s + (−0.104 − 0.0417i)7-s + (−0.0474 − 0.329i)9-s + (−0.0426 − 0.895i)11-s + (−0.654 + 0.0624i)13-s + (0.0989 − 0.687i)15-s + (−0.378 − 1.55i)17-s + (−0.00899 + 0.00360i)19-s + (−0.0576 + 0.0297i)21-s + (1.39 + 0.269i)23-s + (0.186 − 0.408i)25-s + (−0.161 − 0.104i)27-s + (−0.843 + 1.46i)29-s + (1.20 + 0.115i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43132 - 1.28748i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43132 - 1.28748i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-8.13 + 0.910i)T \) |
| good | 5 | \( 1 + (-2.26 + 1.45i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.275 + 0.110i)T + (5.06 + 4.83i)T^{2} \) |
| 11 | \( 1 + (0.141 + 2.97i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (2.35 - 0.225i)T + (12.7 - 2.46i)T^{2} \) |
| 17 | \( 1 + (1.55 + 6.43i)T + (-15.1 + 7.78i)T^{2} \) |
| 19 | \( 1 + (0.0392 - 0.0157i)T + (13.7 - 13.1i)T^{2} \) |
| 23 | \( 1 + (-6.70 - 1.29i)T + (21.3 + 8.54i)T^{2} \) |
| 29 | \( 1 + (4.54 - 7.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.70 - 0.640i)T + (30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (3.58 + 6.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.14 - 1.08i)T + (1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.0911 + 0.0267i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-1.33 - 3.85i)T + (-36.9 + 29.0i)T^{2} \) |
| 53 | \( 1 + (-9.92 + 2.91i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.805 - 1.76i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.0570 - 1.19i)T + (-60.7 - 5.79i)T^{2} \) |
| 71 | \( 1 + (-2.07 + 8.57i)T + (-63.1 - 32.5i)T^{2} \) |
| 73 | \( 1 + (0.527 - 11.0i)T + (-72.6 - 6.93i)T^{2} \) |
| 79 | \( 1 + (0.0496 + 0.0696i)T + (-25.8 + 74.6i)T^{2} \) |
| 83 | \( 1 + (-5.57 - 2.87i)T + (48.1 + 67.6i)T^{2} \) |
| 89 | \( 1 + (-2.48 - 2.86i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (6.97 + 12.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.799955586266823346335313744757, −9.147714226921460119914836430581, −8.617072165983059428418423204713, −7.37306465743608771159426737643, −6.70264426153208185837735190746, −5.49808200372982286352466552357, −4.93889742922178205928973356205, −3.31486527201315115047370013201, −2.30597854038435233398693339164, −0.940493390516585833662049448423,
1.93640863303110472749234032324, 2.77035298820493307842990171179, 4.07976943868187446604755635884, 5.07768905402454054972594263672, 6.16124726468278678731313417514, 6.89317944635392801903955208282, 7.952695494292726311172585206484, 8.911938200957280629511317296991, 9.863844280384891500501848170747, 10.16886860672609765455156442095
