| L(s) = 1 | + (1.36 + 0.365i)2-s + (1.31 − 1.12i)3-s + (1.73 + 0.999i)4-s + (0.342 − 0.939i)5-s + (2.20 − 1.05i)6-s + (1.60 + 0.283i)7-s + (2.00 + 1.99i)8-s + (0.460 − 2.96i)9-s + (0.811 − 1.15i)10-s + (−2.42 + 0.881i)11-s + (3.40 − 0.637i)12-s + (−2.45 + 2.06i)13-s + (2.09 + 0.976i)14-s + (−0.609 − 1.62i)15-s + (2.00 + 3.46i)16-s + (−2.30 + 1.33i)17-s + ⋯ |
| L(s) = 1 | + (0.965 + 0.258i)2-s + (0.759 − 0.650i)3-s + (0.866 + 0.499i)4-s + (0.152 − 0.420i)5-s + (0.901 − 0.432i)6-s + (0.608 + 0.107i)7-s + (0.707 + 0.706i)8-s + (0.153 − 0.988i)9-s + (0.256 − 0.366i)10-s + (−0.730 + 0.265i)11-s + (0.982 − 0.184i)12-s + (−0.681 + 0.572i)13-s + (0.559 + 0.261i)14-s + (−0.157 − 0.418i)15-s + (0.500 + 0.865i)16-s + (−0.559 + 0.323i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.30529 - 0.334276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.30529 - 0.334276i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.365i)T \) |
| 3 | \( 1 + (-1.31 + 1.12i)T \) |
| 5 | \( 1 + (-0.342 + 0.939i)T \) |
| good | 7 | \( 1 + (-1.60 - 0.283i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.42 - 0.881i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.45 - 2.06i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.30 - 1.33i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.766 + 0.442i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.27 + 7.21i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.19 - 5.00i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.65 + 0.644i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.01 - 6.95i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.67 - 4.38i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.35 + 3.71i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.825 + 4.67i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.903iT - 53T^{2} \) |
| 59 | \( 1 + (4.65 + 1.69i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.652 + 3.70i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 - 12.1i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.02 + 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.01 - 3.48i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.44 - 2.91i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.67 - 5.60i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.17 + 1.25i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.0 - 5.10i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.04677484236045025989847768581, −9.889544161003965168100020284676, −8.632066914404732321673961117907, −8.056727076590691734185466544398, −7.09593436373972017137328771240, −6.28912618001469624945546774144, −5.00924458308309352936337658940, −4.23510266505959651201034211232, −2.73727124507423310367014921232, −1.86445130357342578738941569566,
2.10290526276239037937263754041, 2.99400328626472922789643418128, 4.10499361470663945725388654187, 5.05167390819944141791506139904, 5.90133651818987892925152616454, 7.43764730083631325230407977457, 7.88366886023382801871295758573, 9.358976501491549140312367914667, 10.11350495827368203917805361129, 10.93477476236416601976008470525
