| L(s) = 1 | + (−0.987 + 0.156i)2-s + (−1.58 − 0.693i)3-s + (0.951 − 0.309i)4-s + (−0.197 + 2.22i)5-s + (1.67 + 0.436i)6-s + (−1.43 − 1.43i)7-s + (−0.891 + 0.453i)8-s + (2.03 + 2.20i)9-s + (−0.152 − 2.23i)10-s + (−3.22 + 4.44i)11-s + (−1.72 − 0.168i)12-s + (−1.00 + 6.34i)13-s + (1.63 + 1.19i)14-s + (1.85 − 3.39i)15-s + (0.809 − 0.587i)16-s + (−0.101 − 0.199i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (−0.916 − 0.400i)3-s + (0.475 − 0.154i)4-s + (−0.0885 + 0.996i)5-s + (0.684 + 0.178i)6-s + (−0.541 − 0.541i)7-s + (−0.315 + 0.160i)8-s + (0.679 + 0.733i)9-s + (−0.0483 − 0.705i)10-s + (−0.973 + 1.33i)11-s + (−0.497 − 0.0486i)12-s + (−0.278 + 1.75i)13-s + (0.437 + 0.318i)14-s + (0.479 − 0.877i)15-s + (0.202 − 0.146i)16-s + (−0.0246 − 0.0483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.217666 + 0.323059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.217666 + 0.323059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 3 | \( 1 + (1.58 + 0.693i)T \) |
| 5 | \( 1 + (0.197 - 2.22i)T \) |
| good | 7 | \( 1 + (1.43 + 1.43i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.22 - 4.44i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.00 - 6.34i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.101 + 0.199i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.05 - 0.669i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.339 + 2.14i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.33 + 7.19i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.944 - 2.90i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.46 - 1.02i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.896 - 1.23i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.14 - 2.14i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.66 - 2.37i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (5.56 - 10.9i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-4.61 + 3.35i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.55 - 1.85i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.22 + 1.13i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (5.77 - 1.87i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.02 - 0.478i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (11.0 - 3.57i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-9.31 + 4.74i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-7.34 - 5.33i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.09 + 2.14i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.20941254947431278487826283035, −12.05317494581855481192008469439, −11.24182017892530547382584127852, −10.22806338282879890484512253662, −9.638643448729160901788384619166, −7.62704596449076726398530146883, −7.07414108694277132616886048338, −6.18536266567877426138039530057, −4.45240481952677687081039259618, −2.21822682122274921593243262409,
0.51423296140439737326084050146, 3.26267122041513176009071579387, 5.30187900492852077153178818135, 5.84586952774083294510709133256, 7.63412899770909826050472595979, 8.680398539176712691907443228247, 9.705204991208465552763026356219, 10.63669144068544288375479970497, 11.54971800582858136104575617107, 12.65872948562460544841684637751
