L(s) = 1 | + (−0.648 − 1.99i)2-s + (−0.809 − 0.587i)3-s + (−1.94 + 1.41i)4-s + (0.976 − 3.00i)5-s + (−0.648 + 1.99i)6-s + (0.809 − 0.587i)7-s + (0.679 + 0.493i)8-s + (0.309 + 0.951i)9-s − 6.62·10-s + (−0.965 − 3.17i)11-s + 2.40·12-s + (0.657 + 2.02i)13-s + (−1.69 − 1.23i)14-s + (−2.55 + 1.85i)15-s + (−0.939 + 2.89i)16-s + (−1.48 + 4.55i)17-s + ⋯ |
L(s) = 1 | + (−0.458 − 1.41i)2-s + (−0.467 − 0.339i)3-s + (−0.970 + 0.705i)4-s + (0.436 − 1.34i)5-s + (−0.264 + 0.814i)6-s + (0.305 − 0.222i)7-s + (0.240 + 0.174i)8-s + (0.103 + 0.317i)9-s − 2.09·10-s + (−0.291 − 0.956i)11-s + 0.692·12-s + (0.182 + 0.561i)13-s + (−0.453 − 0.329i)14-s + (−0.659 + 0.479i)15-s + (−0.234 + 0.722i)16-s + (−0.359 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.149934 + 0.782694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149934 + 0.782694i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.965 + 3.17i)T \) |
good | 2 | \( 1 + (0.648 + 1.99i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.976 + 3.00i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-0.657 - 2.02i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.48 - 4.55i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.24 + 0.904i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + (-0.775 + 0.563i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.23 + 6.86i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.94 - 1.41i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.215 + 0.156i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 + (8.35 + 6.06i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.292 + 0.899i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.72 + 5.61i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.64 + 8.13i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + (-4.37 + 13.4i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (6.24 - 4.53i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.87 - 8.85i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.20 - 16.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-2.19 - 6.74i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.39872444974633973719161056609, −11.00898644607565888635125774163, −9.837509784837174499415959287843, −8.875583908972978672977517272016, −8.230340266157393320369107715065, −6.40572270229380919495623775096, −5.18350748046861704389517369875, −3.91321379163458574482648273242, −2.03873145263025525450799693367, −0.832891437684013252612664451843,
2.79474261131064137530859781915, 4.86330352092261383880909911642, 5.80657258813701380976155641965, 6.89178198174946815749231113994, 7.34793335572357137380942292262, 8.730622924880327937202782862505, 9.752639419708387166597293373747, 10.60626863931953929073482181905, 11.52572677148230759265228376698, 12.85424671815464609601354678257