L(s) = 1 | + (1.38 + 0.290i)2-s + (1.83 + 0.803i)4-s + (−2.75 − 1.59i)5-s + (−0.944 + 2.47i)7-s + (2.30 + 1.64i)8-s + (−3.35 − 3.00i)10-s + (1.24 + 2.16i)11-s + 6.61·13-s + (−2.02 + 3.14i)14-s + (2.70 + 2.94i)16-s + (2.67 − 1.54i)17-s + (4.40 + 2.54i)19-s + (−3.77 − 5.13i)20-s + (1.10 + 3.35i)22-s + (−2.53 + 4.39i)23-s + ⋯ |
L(s) = 1 | + (0.978 + 0.205i)2-s + (0.915 + 0.401i)4-s + (−1.23 − 0.712i)5-s + (−0.356 + 0.934i)7-s + (0.813 + 0.581i)8-s + (−1.06 − 0.950i)10-s + (0.376 + 0.651i)11-s + 1.83·13-s + (−0.541 + 0.841i)14-s + (0.677 + 0.735i)16-s + (0.649 − 0.374i)17-s + (1.01 + 0.583i)19-s + (−0.843 − 1.14i)20-s + (0.234 + 0.715i)22-s + (−0.529 + 0.916i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.22662 + 1.01550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22662 + 1.01550i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.290i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.944 - 2.47i)T \) |
good | 5 | \( 1 + (2.75 + 1.59i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.24 - 2.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 + (-2.67 + 1.54i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.40 - 2.54i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.53 - 4.39i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.59iT - 29T^{2} \) |
| 31 | \( 1 + (7.31 - 4.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.357 - 0.619i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 + 3.72iT - 43T^{2} \) |
| 47 | \( 1 + (-2.51 + 4.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.21 - 4.16i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.997 + 1.72i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.00277 - 0.00160i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + (-0.957 - 1.65i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.11 - 3.52i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.93T + 83T^{2} \) |
| 89 | \( 1 + (0.482 + 0.278i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.127T + 97T^{2} \) |
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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
−10.83142958087800048868387165861, −9.442954463779181082518602052352, −8.561806739404388293563144029853, −7.80503104927118378158275469135, −6.93487062132484912354391281781, −5.76451668138371984370699697480, −5.16488059663790687470996831760, −3.78868582829250742420714417317, −3.47448501887130927421995925666, −1.61324995025645879956972378108,
1.06115730518947303035949287163, 3.15464857508528308504928428749, 3.64242651231172826658749938108, 4.38772221946942792723655202792, 5.93558047092979111361015389649, 6.55698648993908169834836098085, 7.51530810220588954296754577944, 8.172678099750389313835799072028, 9.608517676099800127467160792156, 10.74690477670629597788278926430