| L(s) = 1 | + (0.978 + 0.207i)2-s + (1.25 + 2.17i)3-s + (0.913 + 0.406i)4-s + (0.134 + 1.28i)5-s + (0.776 + 2.39i)6-s + (1.17 + 2.36i)7-s + (0.809 + 0.587i)8-s + (−1.65 + 2.87i)9-s + (−0.134 + 1.28i)10-s + (0.456 − 4.34i)11-s + (0.262 + 2.49i)12-s + (−0.797 − 2.45i)13-s + (0.658 + 2.56i)14-s + (−2.62 + 1.90i)15-s + (0.669 + 0.743i)16-s + (0.414 − 3.94i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (0.725 + 1.25i)3-s + (0.456 + 0.203i)4-s + (0.0603 + 0.574i)5-s + (0.317 + 0.975i)6-s + (0.444 + 0.895i)7-s + (0.286 + 0.207i)8-s + (−0.552 + 0.957i)9-s + (−0.0426 + 0.405i)10-s + (0.137 − 1.31i)11-s + (0.0758 + 0.721i)12-s + (−0.221 − 0.680i)13-s + (0.175 + 0.684i)14-s + (−0.677 + 0.492i)15-s + (0.167 + 0.185i)16-s + (0.100 − 0.955i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88943 + 2.11948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88943 + 2.11948i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.17 - 2.36i)T \) |
| 41 | \( 1 + (-4.35 - 4.69i)T \) |
| good | 3 | \( 1 + (-1.25 - 2.17i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.134 - 1.28i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.456 + 4.34i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.797 + 2.45i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.414 + 3.94i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.743 - 0.825i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (8.32 + 1.76i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (6.83 - 4.96i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.548 + 5.22i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.0107 + 0.102i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (1.57 + 4.85i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-9.87 - 2.09i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.69 - 1.19i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (2.90 - 3.22i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.71 + 3.02i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.29 - 1.02i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (9.97 + 7.24i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.17 - 5.50i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.75 - 9.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 + (5.51 + 6.12i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-6.36 + 4.62i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.95329741031260743401208416169, −10.14938794480393167093692705348, −9.177294623123105501967955995158, −8.433194529258122424093076162299, −7.51829684066626384685265001237, −6.03060720303315473466261683765, −5.38947764281129112082612606429, −4.24774949906742285984742984613, −3.24761948907939919426547268090, −2.53802290123095257582939425657,
1.46780191006504652896669562295, 2.16021923211037783601898573826, 3.85400259912394461590080308820, 4.64532649186049523692065276884, 6.01291067605180480129150265774, 7.13637482172686392777716539928, 7.53360562022495223066592425049, 8.505383059036281911335253774377, 9.612641241801704975853351393269, 10.55889552286674986732978968459
