L(s) = 1 | − 2·2-s + (−0.5 + 0.866i)3-s + 4·4-s + (−4.5 + 2.59i)5-s + (1 − 1.73i)6-s + (−1 + 6.92i)7-s − 8·8-s + (4 + 6.92i)9-s + (9 − 5.19i)10-s + (−8.5 + 14.7i)11-s + (−2 + 3.46i)12-s − 13.8i·13-s + (2 − 13.8i)14-s − 5.19i·15-s + 16·16-s + (12.5 − 21.6i)17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.166 + 0.288i)3-s + 4-s + (−0.900 + 0.519i)5-s + (0.166 − 0.288i)6-s + (−0.142 + 0.989i)7-s − 8-s + (0.444 + 0.769i)9-s + (0.900 − 0.519i)10-s + (−0.772 + 1.33i)11-s + (−0.166 + 0.288i)12-s − 1.06i·13-s + (0.142 − 0.989i)14-s − 0.346i·15-s + 16-s + (0.735 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.319031 + 0.440366i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.319031 + 0.440366i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 + (1 - 6.92i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.5 - 14.7i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + (-12.5 + 21.6i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-4.5 + 2.59i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 13.8iT - 841T^{2} \) |
| 31 | \( 1 + (-28.5 - 16.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-7.5 + 4.33i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 26T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14T + 1.84e3T^{2} \) |
| 47 | \( 1 + (43.5 - 25.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-79.5 - 45.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-27.5 + 47.6i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-19.5 + 11.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (8.5 - 14.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + (59.5 - 103. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-64.5 + 37.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 110T + 6.88e3T^{2} \) |
| 89 | \( 1 + (35.5 + 61.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 22T + 9.40e3T^{2} \) |
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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
−15.64162963855443003717631900362, −14.84216110503369131336556574266, −12.68728471432642379838181288645, −11.73449471971366651011079544095, −10.54188698329264581462303970459, −9.669941328944072545979456327052, −8.005478117488796877493615808269, −7.25559595834741959298588121959, −5.26638285427959323095025978372, −2.76162049414574367655546538429,
0.75302290757209116481621355117, 3.78923785712948819789823619625, 6.27667829687070658451074276547, 7.57869474807904611022942158324, 8.533955750148234926553709147054, 9.978493070682500060836092791285, 11.18828228464196321820902704917, 12.11485452206431359743471847798, 13.39609390185370597348341429945, 15.03693907958629444804486106852