L(s) = 1 | + (1.07 − 0.917i)2-s + (−0.0574 − 1.73i)3-s + (0.317 − 1.97i)4-s + (0.461 − 0.266i)5-s + (−1.64 − 1.81i)6-s + (0.489 + 2.60i)7-s + (−1.46 − 2.41i)8-s + (−2.99 + 0.198i)9-s + (0.252 − 0.710i)10-s + (−2.28 + 3.96i)11-s + (−3.43 − 0.435i)12-s + 4.97·13-s + (2.91 + 2.35i)14-s + (−0.487 − 0.783i)15-s + (−3.79 − 1.25i)16-s + (2.16 − 3.74i)17-s + ⋯ |
L(s) = 1 | + (0.761 − 0.648i)2-s + (−0.0331 − 0.999i)3-s + (0.158 − 0.987i)4-s + (0.206 − 0.119i)5-s + (−0.673 − 0.739i)6-s + (0.184 + 0.982i)7-s + (−0.519 − 0.854i)8-s + (−0.997 + 0.0663i)9-s + (0.0798 − 0.224i)10-s + (−0.689 + 1.19i)11-s + (−0.992 − 0.125i)12-s + 1.38·13-s + (0.778 + 0.628i)14-s + (−0.125 − 0.202i)15-s + (−0.949 − 0.313i)16-s + (0.524 − 0.907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07664 - 1.25770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07664 - 1.25770i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.07 + 0.917i)T \) |
| 3 | \( 1 + (0.0574 + 1.73i)T \) |
| 7 | \( 1 + (-0.489 - 2.60i)T \) |
good | 5 | \( 1 + (-0.461 + 0.266i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.28 - 3.96i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + (-2.16 + 3.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.921 + 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.103 - 0.0596i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 + (1.93 + 1.11i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.02 - 4.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 + 1.87iT - 43T^{2} \) |
| 47 | \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.29 + 3.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.71 + 3.29i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.07 + 1.86i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.4 + 6.05i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.20iT - 71T^{2} \) |
| 73 | \( 1 + (-8.35 - 4.82i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0228 + 0.0396i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.86iT - 83T^{2} \) |
| 89 | \( 1 + (8.23 + 14.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.18iT - 97T^{2} \) |
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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
−12.48858267911812478132648830213, −11.85875857686233172318954170528, −10.89110380570261216381292345787, −9.586528964309799800252825392667, −8.451731252455355338231100529604, −7.02943276274255629211876095094, −5.88595628478146647846752194026, −4.96812585601088954868736165919, −2.97028377650107347997880534057, −1.71407151699329298752400282239,
3.29736191654829264380649068222, 4.14939917035060803925685780030, 5.54100643170657471559029363189, 6.36367044823122065970557569331, 8.037362741598641283048095552914, 8.667189834511790552293927383610, 10.44805519656785970648875612779, 10.85888865933908091922407801962, 12.13962141148738200145995752897, 13.62893009069681358704542395965