L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (2.02 − 0.951i)5-s + 3.77·7-s + (−0.309 − 0.951i)8-s + (1.07 − 1.95i)10-s + (−3.05 + 2.21i)11-s + (−2.56 − 1.86i)13-s + (3.05 − 2.21i)14-s + (−0.809 − 0.587i)16-s + (0.430 + 1.32i)17-s + (1.20 + 3.72i)19-s + (−0.279 − 2.21i)20-s + (−1.16 + 3.58i)22-s + (0.720 − 0.523i)23-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.905 − 0.425i)5-s + 1.42·7-s + (−0.109 − 0.336i)8-s + (0.340 − 0.619i)10-s + (−0.920 + 0.668i)11-s + (−0.712 − 0.517i)13-s + (0.816 − 0.592i)14-s + (−0.202 − 0.146i)16-s + (0.104 + 0.321i)17-s + (0.277 + 0.853i)19-s + (−0.0624 − 0.496i)20-s + (−0.248 + 0.765i)22-s + (0.150 − 0.109i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09678 - 0.987958i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09678 - 0.987958i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.02 + 0.951i)T \) |
good | 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + (3.05 - 2.21i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.56 + 1.86i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.430 - 1.32i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.20 - 3.72i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.720 + 0.523i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0152 - 0.0468i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.70 + 4.14i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.20 + 0.875i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 + (1.16 - 3.58i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.58 - 11.0i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (0.558 + 0.405i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.38 - 6.08i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.73 - 14.5i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.06 + 6.36i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.18 - 3.03i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.558 - 1.71i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.08 - 9.47i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 8.52i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.0278 - 0.0857i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.85323658544293765542849335972, −10.28893561850749627512963919252, −9.370887921141188496289747615361, −8.169757562543099033392265656728, −7.37479680591133865416342545137, −5.78985093496032736759548337778, −5.19147893142958680608662909544, −4.33888315945612470963091240551, −2.55714357764392171769783156931, −1.57351132688070417888829746402,
1.95368324158015536629900332659, 3.13541108751778977217213067803, 4.95073826756198367546128255647, 5.19785598001652967991005606492, 6.53209068885932105836208812156, 7.42918751408149043500280998632, 8.341623189707837390887934542348, 9.342326090366786326984383804464, 10.54237208538192657119467919167, 11.19929387020706111160429175357