| L(s) = 1 | + (1.24 + 0.661i)2-s + (0.540 + 0.841i)3-s + (1.12 + 1.65i)4-s + (−2.21 + 1.01i)5-s + (0.118 + 1.40i)6-s + (−3.93 + 1.15i)7-s + (0.309 + 2.81i)8-s + (−0.415 + 0.909i)9-s + (−3.43 − 0.201i)10-s + (1.69 − 1.46i)11-s + (−0.784 + 1.83i)12-s + (0.366 − 1.24i)13-s + (−5.68 − 1.16i)14-s + (−2.04 − 1.31i)15-s + (−1.47 + 3.71i)16-s + (−0.326 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (0.883 + 0.468i)2-s + (0.312 + 0.485i)3-s + (0.561 + 0.827i)4-s + (−0.989 + 0.451i)5-s + (0.0485 + 0.575i)6-s + (−1.48 + 0.436i)7-s + (0.109 + 0.993i)8-s + (−0.138 + 0.303i)9-s + (−1.08 − 0.0637i)10-s + (0.510 − 0.442i)11-s + (−0.226 + 0.531i)12-s + (0.101 − 0.346i)13-s + (−1.51 − 0.310i)14-s + (−0.528 − 0.339i)15-s + (−0.368 + 0.929i)16-s + (−0.0792 + 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.397787 + 1.68145i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.397787 + 1.68145i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 - 0.661i)T \) |
| 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 23 | \( 1 + (4.34 - 2.03i)T \) |
| good | 5 | \( 1 + (2.21 - 1.01i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.93 - 1.15i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 1.46i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.366 + 1.24i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.326 - 2.27i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-3.05 + 0.439i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.81 - 0.979i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.892 - 0.573i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.62 + 0.744i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.91 - 10.7i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.53 - 2.39i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 + (-2.57 - 8.76i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.65 + 9.03i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.08 + 4.79i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (3.69 + 3.20i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (3.08 - 3.55i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.00 + 6.96i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (2.34 + 0.687i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (6.16 + 2.81i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.684 + 0.439i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.23 + 7.08i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.37259745800773754865567091219, −10.36339668591292995073595787752, −9.318402348061372999598032925324, −8.335019951214525383154119128934, −7.48405059524863325804137518500, −6.45861616167679058742124593800, −5.77047696879847078035807673432, −4.32149725682443746055283743645, −3.46189249004792853220735082170, −2.88348469004561249799814608692,
0.71693307572177405166914394180, 2.55110508459223540682025073412, 3.72398842905308549998913167971, 4.30123442957002529672549695030, 5.79853342950648381359344527250, 6.82085196247200538043342028221, 7.36037837354581801259957576770, 8.734454702597261040603304212351, 9.692253201462797688352466092207, 10.43523081533737120629302423082
