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
−10.74690477670629597788278926430, −9.608517676099800127467160792156, −8.172678099750389313835799072028, −7.51530810220588954296754577944, −6.55698648993908169834836098085, −5.93558047092979111361015389649, −4.38772221946942792723655202792, −3.64242651231172826658749938108, −3.15464857508528308504928428749, −1.06115730518947303035949287163,
1.61324995025645879956972378108, 3.47448501887130927421995925666, 3.78868582829250742420714417317, 5.16488059663790687470996831760, 5.76451668138371984370699697480, 6.93487062132484912354391281781, 7.80503104927118378158275469135, 8.561806739404388293563144029853, 9.442954463779181082518602052352, 10.83142958087800048868387165861