L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.448 + 1.67i)3-s + (−0.866 − 0.499i)4-s + (1.50 + 0.866i)6-s + (3.34 + 0.896i)7-s + (−0.707 + 0.707i)8-s + (−2.59 − 1.50i)9-s + (−1.5 + 0.866i)11-s + (1.22 − 1.22i)12-s + (3.34 − 0.896i)13-s + (1.73 − 3.00i)14-s + (0.500 + 0.866i)16-s + (2.12 + 2.12i)17-s + (−2.12 + 2.12i)18-s + 7i·19-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.258 + 0.965i)3-s + (−0.433 − 0.249i)4-s + (0.612 + 0.353i)6-s + (1.26 + 0.338i)7-s + (−0.249 + 0.249i)8-s + (−0.866 − 0.5i)9-s + (−0.452 + 0.261i)11-s + (0.353 − 0.353i)12-s + (0.928 − 0.248i)13-s + (0.462 − 0.801i)14-s + (0.125 + 0.216i)16-s + (0.514 + 0.514i)17-s + (−0.499 + 0.499i)18-s + 1.60i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43441 + 0.340633i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43441 + 0.340633i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.448 - 1.67i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.34 - 0.896i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.12 - 2.12i)T + 17iT^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 + (-1.55 - 5.79i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 3i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \) |
| 41 | \( 1 + (10.5 + 6.06i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.34 - 5.01i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.55 - 5.79i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + (-6.06 + 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.448 - 1.67i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (6.12 + 6.12i)T + 73iT^{2} \) |
| 79 | \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.5 + 3.10i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + (8.36 + 2.24i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.17606108287518458553558721719, −10.37767298588665121651882719550, −9.658343548661989121547606263563, −8.505899995404746537013381861478, −7.903637250055323564531717600085, −5.94123440525801380684108002596, −5.31461083795878712010925715524, −4.26040047818561177554415923432, −3.29546566246711301907190401366, −1.63326013519296556944833134473,
1.05130341901194963155795700792, 2.78971947913349366466335164194, 4.60076058180777853225636636410, 5.32816062475016421507500726311, 6.56185425706778830226047317996, 7.18026300851830165049019262627, 8.412565056350852589718923364169, 8.496375185259444376499577311585, 10.27308056529825518338415465609, 11.31093143937489221714975118558