| L(s) = 1 | + (−0.960 + 1.03i)2-s + (−0.755 + 0.654i)3-s + (−0.153 − 1.99i)4-s + (0.708 − 0.101i)5-s + (0.0466 − 1.41i)6-s + (−1.58 + 3.47i)7-s + (2.21 + 1.75i)8-s + (0.142 − 0.989i)9-s + (−0.574 + 0.832i)10-s + (−0.408 + 1.39i)11-s + (1.42 + 1.40i)12-s + (5.37 − 2.45i)13-s + (−2.08 − 4.98i)14-s + (−0.468 + 0.540i)15-s + (−3.95 + 0.612i)16-s + (−2.62 − 1.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.679 + 0.733i)2-s + (−0.436 + 0.378i)3-s + (−0.0767 − 0.997i)4-s + (0.316 − 0.0455i)5-s + (0.0190 − 0.577i)6-s + (−0.600 + 1.31i)7-s + (0.783 + 0.621i)8-s + (0.0474 − 0.329i)9-s + (−0.181 + 0.263i)10-s + (−0.123 + 0.419i)11-s + (0.410 + 0.406i)12-s + (1.48 − 0.680i)13-s + (−0.556 − 1.33i)14-s + (−0.120 + 0.139i)15-s + (−0.988 + 0.153i)16-s + (−0.637 − 0.409i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0660002 + 0.627425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0660002 + 0.627425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.960 - 1.03i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (4.77 + 0.419i)T \) |
| good | 5 | \( 1 + (-0.708 + 0.101i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.58 - 3.47i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.408 - 1.39i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.37 + 2.45i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.62 + 1.68i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.06 - 6.33i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (4.55 - 7.08i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.311 + 0.359i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.60 + 0.805i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.28 - 8.93i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 1.82i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 9.11T + 47T^{2} \) |
| 53 | \( 1 + (4.16 + 1.90i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.26 + 0.578i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.93 - 3.40i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (3.26 + 11.1i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (13.1 - 3.87i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.18 + 4.61i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.49 - 12.0i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-11.5 - 1.66i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (3.74 + 4.32i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.61 - 11.2i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.98810662168012556711260965078, −10.02911036791188561004282092477, −9.446903005763003610269604867016, −8.636179067296837636610368096504, −7.75277528770764561387934997270, −6.39049825658236001954228346842, −5.85160877181583016680744413817, −5.15771380718167890523612658009, −3.50306129262704672306113949240, −1.74444876279046779600752306198,
0.48325556051793370663464955103, 1.86997823145931357105961466432, 3.47759027893010463865026495911, 4.30390625349304739032279791277, 6.03678824028460977196948943203, 6.86450502988614022974891426229, 7.71069005500912156902926022919, 8.756487655313481717407807513384, 9.656181010207931663167951665996, 10.46587666552693352991805475432
