L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.189 − 2.63i)7-s − 0.999i·8-s + (−2.55 − 1.47i)11-s − 3.93i·13-s + (−1.48 − 2.19i)14-s + (−0.5 − 0.866i)16-s + (−0.199 + 0.346i)17-s + (0.0305 − 0.0176i)19-s − 2.94·22-s + (3.23 − 1.86i)23-s + (−1.96 − 3.40i)26-s + (−2.38 − 1.15i)28-s + 8.89i·29-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0716 − 0.997i)7-s − 0.353i·8-s + (−0.769 − 0.444i)11-s − 1.09i·13-s + (−0.396 − 0.585i)14-s + (−0.125 − 0.216i)16-s + (−0.0484 + 0.0839i)17-s + (0.00700 − 0.00404i)19-s − 0.628·22-s + (0.673 − 0.389i)23-s + (−0.385 − 0.667i)26-s + (−0.449 − 0.218i)28-s + 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.557685623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.557685623\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.189 + 2.63i)T \) |
good | 11 | \( 1 + (2.55 + 1.47i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.93iT - 13T^{2} \) |
| 17 | \( 1 + (0.199 - 0.346i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0305 + 0.0176i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-0.717 - 0.414i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.96 + 6.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 + (2.90 + 5.02i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.72 + 2.14i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.78 - 4.82i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.97 - 5.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.25 - 10.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.93iT - 71T^{2} \) |
| 73 | \( 1 + (-0.297 - 0.171i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.15 + 7.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + (-3.08 - 5.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
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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
−8.294630622782402262875312649870, −7.40918347353374272868713145835, −6.89070404813788554332598132649, −5.83624122421751621383979549828, −5.19870449834161559299212887900, −4.43004799157683683118238136598, −3.38553663913116955537356845714, −2.92616631411907260065898799591, −1.53602425015116208156085436635, −0.36426027656358802423075337262,
1.80075146859170239376575391977, 2.62606557595000936560621918606, 3.52605672019096578168114958964, 4.70860975191696333665313116928, 5.03732469405460327423565736245, 6.14263400695913073967087902883, 6.53432200817726472159501178911, 7.58403734467139884556234286361, 8.111663271499211310496404019194, 9.073001953704082155154597454111