L(s) = 1 | + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (0.621 − 0.358i)5-s + 2.44i·6-s + (−2.62 − 0.358i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (0.878 + 0.507i)10-s + (2.91 − 5.04i)11-s + (−2.99 + 1.73i)12-s + (−1.41 − 3.46i)14-s + 1.24·15-s + (−2.00 − 3.46i)16-s + (−2.12 + 3.67i)18-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)2-s + (0.866 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.277 − 0.160i)5-s + 0.999i·6-s + (−0.990 − 0.135i)7-s − 0.999·8-s + (0.5 + 0.866i)9-s + (0.277 + 0.160i)10-s + (0.878 − 1.52i)11-s + (−0.866 + 0.500i)12-s + (−0.377 − 0.925i)14-s + 0.320·15-s + (−0.500 − 0.866i)16-s + (−0.499 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00925 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00925 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21081 + 1.19966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21081 + 1.19966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-0.621 + 0.358i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.91 + 5.04i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + (9.62 + 5.55i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.03 - 3.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (12.9 + 7.49i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.86 - 15.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.06iT - 97T^{2} \) |
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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
−13.49644490305673902599548191684, −12.46008047148235720539754253377, −11.03318692728727745205172008456, −9.535930006267896807775903128464, −8.984240680934966648402970104489, −7.916220613973956293942346581565, −6.62971602430470037355028976819, −5.58607104680017025375238457088, −4.00069423234993990684982530594, −3.12843816523757430837690512104,
1.85612458809570943607386661198, 3.15682033607213618155360099238, 4.39130656411964321773884536855, 6.20966689603491368244029360232, 7.12005730994234531843561402131, 8.862718452280285750200296245631, 9.607895928041710259448305815718, 10.36027783826117806662156251383, 12.07795045395855681803578194940, 12.46618357453874285774754724401