L(s) = 1 | + (0.648 + 1.25i)2-s + (−1.15 + 1.63i)4-s + (1.02 − 1.78i)5-s + (1.24 − 2.33i)7-s + (−2.80 − 0.396i)8-s + (2.90 + 0.136i)10-s + (5.24 − 3.03i)11-s − 1.77i·13-s + (3.74 + 0.0525i)14-s + (−1.31 − 3.77i)16-s + (0.786 − 0.453i)17-s + (−3.37 + 5.84i)19-s + (1.71 + 3.73i)20-s + (7.21 + 4.62i)22-s + (0.351 − 0.608i)23-s + ⋯ |
L(s) = 1 | + (0.458 + 0.888i)2-s + (−0.578 + 0.815i)4-s + (0.459 − 0.796i)5-s + (0.471 − 0.882i)7-s + (−0.990 − 0.140i)8-s + (0.918 + 0.0431i)10-s + (1.58 − 0.913i)11-s − 0.493i·13-s + (0.999 + 0.0140i)14-s + (−0.329 − 0.944i)16-s + (0.190 − 0.110i)17-s + (−0.774 + 1.34i)19-s + (0.383 + 0.835i)20-s + (1.53 + 0.987i)22-s + (0.0732 − 0.126i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91809 + 0.376025i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91809 + 0.376025i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.648 - 1.25i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.24 + 2.33i)T \) |
good | 5 | \( 1 + (-1.02 + 1.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.24 + 3.03i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.77iT - 13T^{2} \) |
| 17 | \( 1 + (-0.786 + 0.453i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.37 - 5.84i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.351 + 0.608i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + (7.31 - 4.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.07 - 3.50i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 11.6iT - 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 + (4.42 - 7.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.31 - 5.74i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.11 - 0.645i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.95 - 4.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.562 + 0.974i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + (-2.08 - 3.61i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (11.6 + 6.71i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.71iT - 83T^{2} \) |
| 89 | \( 1 + (4.08 + 2.35i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−11.05347655828264107800293151326, −9.907934832079715254726931087342, −8.848536135491650487859638698872, −8.344232618476817272112872928470, −7.24383762150359695962832648585, −6.26205141276111380336572400382, −5.44621319751201262909470382153, −4.33553865310867890314586922234, −3.52480183223207983565736100964, −1.21958548899971082391143522992,
1.74956469596153040838839839363, 2.63715950325397131850700240636, 4.04408682719901278978590489143, 4.97175151947224035456092360018, 6.24090526430481135337834346411, 6.83499216682404202678824176274, 8.543187453612036915196153414886, 9.372080318946564096781463671953, 9.957089954474190612627077653091, 11.23327222365702887444940734158