| L(s) = 1 | + (−1.12 − 2.20i)2-s + (−1.64 + 0.550i)3-s + (−2.43 + 3.35i)4-s + (−2.18 + 0.479i)5-s + (3.06 + 3.00i)6-s + (−1.56 − 1.56i)7-s + (5.24 + 0.829i)8-s + (2.39 − 1.80i)9-s + (3.51 + 4.28i)10-s + (−2.96 − 0.963i)11-s + (2.15 − 6.84i)12-s + (−1.61 − 0.821i)13-s + (−1.69 + 5.21i)14-s + (3.32 − 1.99i)15-s + (−1.50 − 4.62i)16-s + (−0.290 + 1.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.795 − 1.56i)2-s + (−0.948 + 0.317i)3-s + (−1.21 + 1.67i)4-s + (−0.976 + 0.214i)5-s + (1.25 + 1.22i)6-s + (−0.591 − 0.591i)7-s + (1.85 + 0.293i)8-s + (0.797 − 0.602i)9-s + (1.11 + 1.35i)10-s + (−0.894 − 0.290i)11-s + (0.621 − 1.97i)12-s + (−0.447 − 0.227i)13-s + (−0.452 + 1.39i)14-s + (0.857 − 0.514i)15-s + (−0.375 − 1.15i)16-s + (−0.0704 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0503226 + 0.107245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0503226 + 0.107245i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.64 - 0.550i)T \) |
| 5 | \( 1 + (2.18 - 0.479i)T \) |
| good | 2 | \( 1 + (1.12 + 2.20i)T + (-1.17 + 1.61i)T^{2} \) |
| 7 | \( 1 + (1.56 + 1.56i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.96 + 0.963i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.61 + 0.821i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.290 - 1.83i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (3.62 + 4.98i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.38 + 1.72i)T + (13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-1.02 - 0.743i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.59 - 1.88i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.20 - 4.32i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.49 + 1.13i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.478 + 0.478i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.8 - 1.87i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (0.590 + 3.72i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (3.54 + 10.9i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.39 - 4.29i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.12 + 0.336i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (2.82 - 3.88i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.61 - 5.12i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (0.567 - 0.781i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.118 + 0.0187i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (3.39 - 10.4i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.60 + 16.4i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−13.06349820152740353020659548333, −12.49698019939017163883236510200, −11.23783242223008526359377386587, −10.76463029998037945993618670093, −9.854439301631481830138094260574, −8.427817236387371833602603215051, −6.95318895905335868984830790406, −4.59468909109578057490697890200, −3.21501499151319557323312458264, −0.21915368799849769860659765837,
4.83712797999893565336478935951, 5.99349682424034997860048418164, 7.18228019473913251267123344382, 7.993951078785933556483615674608, 9.338601459244012863944220357977, 10.60123620533649373402447368181, 12.06676665779988176135708723963, 13.04153398973744584354705684762, 14.74867796337539598223182406890, 15.64007030744557448881915801823
