| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−0.268 − 0.464i)3-s + (0.913 + 0.406i)4-s + (−0.301 − 2.87i)5-s + (0.165 + 0.509i)6-s + (−0.844 + 2.50i)7-s + (−0.809 − 0.587i)8-s + (1.35 − 2.34i)9-s + (−0.301 + 2.87i)10-s + (0.151 − 1.44i)11-s + (−0.0560 − 0.533i)12-s + (0.739 + 2.27i)13-s + (1.34 − 2.27i)14-s + (−1.25 + 0.910i)15-s + (0.669 + 0.743i)16-s + (0.751 − 7.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.154 − 0.268i)3-s + (0.456 + 0.203i)4-s + (−0.135 − 1.28i)5-s + (0.0676 + 0.208i)6-s + (−0.319 + 0.947i)7-s + (−0.286 − 0.207i)8-s + (0.452 − 0.783i)9-s + (−0.0954 + 0.908i)10-s + (0.0458 − 0.435i)11-s + (−0.0161 − 0.153i)12-s + (0.205 + 0.631i)13-s + (0.359 − 0.608i)14-s + (−0.323 + 0.235i)15-s + (0.167 + 0.185i)16-s + (0.182 − 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.234733 - 0.663621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.234733 - 0.663621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (0.844 - 2.50i)T \) |
| 41 | \( 1 + (-3.22 - 5.52i)T \) |
| good | 3 | \( 1 + (0.268 + 0.464i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.301 + 2.87i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.151 + 1.44i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.739 - 2.27i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.751 + 7.14i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.416 - 0.462i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (8.53 + 1.81i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-0.258 + 0.187i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.674 - 6.41i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.757 + 7.21i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (3.77 + 11.6i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (9.07 + 1.92i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.166 - 0.0742i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.80 - 2.00i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (3.55 + 3.94i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.30 - 1.91i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (6.76 + 4.91i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.414 - 0.718i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.70 + 6.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + (-4.25 - 4.72i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 7.30i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.11522552329380781150922969658, −9.227081514376934244216664186884, −8.914901969522361150770015358221, −7.933614041989527905531047985035, −6.81553634281458775041442374693, −5.90044163391504634487455043503, −4.83167390937290218994802944713, −3.48317045836315612292701490341, −1.91930681417515927390854762683, −0.50387812746249788054215700130,
1.82812286756396519315282861692, 3.32877616203905968628095214238, 4.32421244470630223673289448364, 5.94157544743323538831028542797, 6.65615826075057452294314667098, 7.72927853042138739253441258913, 8.024050132343351287527918018916, 9.785653990425180688414639102504, 10.19021452639140940749798571596, 10.74923707521430206128575471906
