L(s) = 1 | + (0.563 + 0.826i)2-s + (0.634 − 1.61i)3-s + (−0.365 + 0.930i)4-s + (3.89 + 1.20i)5-s + (1.68 − 0.383i)6-s + (0.170 + 2.64i)7-s + (−0.974 + 0.222i)8-s + (−2.19 − 2.04i)9-s + (1.20 + 3.89i)10-s + (−4.55 + 0.341i)11-s + (1.26 + 1.17i)12-s + (1.44 − 2.99i)13-s + (−2.08 + 1.62i)14-s + (4.40 − 5.51i)15-s + (−0.733 − 0.680i)16-s + (−0.806 − 0.121i)17-s + ⋯ |
L(s) = 1 | + (0.398 + 0.584i)2-s + (0.366 − 0.930i)3-s + (−0.182 + 0.465i)4-s + (1.74 + 0.537i)5-s + (0.689 − 0.156i)6-s + (0.0645 + 0.997i)7-s + (−0.344 + 0.0786i)8-s + (−0.731 − 0.681i)9-s + (0.379 + 1.23i)10-s + (−1.37 + 0.102i)11-s + (0.366 + 0.340i)12-s + (0.400 − 0.831i)13-s + (−0.557 + 0.435i)14-s + (1.13 − 1.42i)15-s + (−0.183 − 0.170i)16-s + (−0.195 − 0.0294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97998 + 0.496619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97998 + 0.496619i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.563 - 0.826i)T \) |
| 3 | \( 1 + (-0.634 + 1.61i)T \) |
| 7 | \( 1 + (-0.170 - 2.64i)T \) |
good | 5 | \( 1 + (-3.89 - 1.20i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (4.55 - 0.341i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-1.44 + 2.99i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.806 + 0.121i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.568 - 0.327i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.306 + 2.03i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.33 + 1.85i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.95 + 2.28i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.44 + 3.69i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (2.36 + 10.3i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.407 + 1.78i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (10.8 - 7.37i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-6.90 - 2.71i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (9.88 - 3.04i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-9.28 + 3.64i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-4.09 - 7.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.00 - 3.19i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.33 - 9.29i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (0.703 - 1.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.4 + 5.97i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.501 - 6.69i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−12.28455974369916511990396460921, −10.89179528869541753306400679967, −9.830513525875356253322136844311, −8.810759166737797577348324441993, −7.925450974814859435809372991640, −6.77621535594776157596211552072, −5.81182081024182197092081467537, −5.41183527428152341847979386394, −2.91979536256857586894514931104, −2.18705669287700144544896411990,
1.80154032172172566931877281252, 3.14514759521738886607365747364, 4.63969207334389447320959329533, 5.26713142428880263636110754290, 6.45788513154491854046017809082, 8.197978845030156868518054664604, 9.247491701460976591773295523881, 10.02670377058878849783165358642, 10.47530574492700085330510498108, 11.45863306031003890415956059239