| L(s) = 1 | + (2.60 + 1.50i)2-s + (−0.740 + 2.90i)3-s + (2.52 + 4.36i)4-s + (−1.93 − 1.11i)5-s + (−6.29 + 6.45i)6-s + (−0.494 + 6.98i)7-s + 3.12i·8-s + (−7.90 − 4.30i)9-s + (−3.36 − 5.82i)10-s + (16.3 − 9.44i)11-s + (−14.5 + 4.09i)12-s + 17.2·13-s + (−11.7 + 17.4i)14-s + (4.68 − 4.80i)15-s + (5.37 − 9.31i)16-s + (−21.7 + 12.5i)17-s + ⋯ |
| L(s) = 1 | + (1.30 + 0.751i)2-s + (−0.246 + 0.969i)3-s + (0.630 + 1.09i)4-s + (−0.387 − 0.223i)5-s + (−1.04 + 1.07i)6-s + (−0.0706 + 0.997i)7-s + 0.391i·8-s + (−0.878 − 0.478i)9-s + (−0.336 − 0.582i)10-s + (1.48 − 0.859i)11-s + (−1.21 + 0.341i)12-s + 1.32·13-s + (−0.841 + 1.24i)14-s + (0.312 − 0.320i)15-s + (0.336 − 0.582i)16-s + (−1.28 + 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38334 + 1.81257i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38334 + 1.81257i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.740 - 2.90i)T \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (0.494 - 6.98i)T \) |
| good | 2 | \( 1 + (-2.60 - 1.50i)T + (2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (-16.3 + 9.44i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 17.2T + 169T^{2} \) |
| 17 | \( 1 + (21.7 - 12.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.28 - 2.22i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.196 + 0.113i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 11.7iT - 841T^{2} \) |
| 31 | \( 1 + (26.5 + 45.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.41 - 9.38i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 4.80iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.49T + 1.84e3T^{2} \) |
| 47 | \( 1 + (0.114 + 0.0663i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (28.3 - 16.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (35.6 - 20.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (3.00 - 5.20i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (48.0 + 83.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-6.02 - 10.4i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (43.6 - 75.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 34.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-14.2 - 8.23i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 48.3T + 9.40e3T^{2} \) |
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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
−14.03917898519178940188568236315, −12.98056601662489918470959857956, −11.77813349314107959488448202856, −11.12633842030532579974785822427, −9.264579904895062864602057074842, −8.462238696206047242830228968341, −6.32919231805800183686973254223, −5.86157604051549097670115926135, −4.34574519205917740437299605092, −3.54472207189594378410623180594,
1.57689889156026864747022789022, 3.48503668848066565185559127371, 4.63116524980907027544502506276, 6.36794179238287455983269246506, 7.14381230997454321296430695749, 8.817694167633357003907163556610, 10.76595786049376703475734181622, 11.37791868488337780431676877715, 12.28108354385016584792400427334, 13.19982846643111426497670734169
