L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·6-s + (−1.62 + 2.82i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 5.25·11-s + (0.499 + 0.866i)12-s + (−3.16 − 1.82i)13-s + 3.25i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (−1.95 + 1.12i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (−0.615 + 1.06i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.58·11-s + (0.144 + 0.249i)12-s + (−0.878 − 0.507i)13-s + 0.870i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (−0.474 + 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3399101048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3399101048\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 37 | \( 1 + (5.27 + 3.03i)T \) |
good | 7 | \( 1 + (1.62 - 2.82i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 1.82i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.95 - 1.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.21 + 1.85i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.92iT - 23T^{2} \) |
| 29 | \( 1 + 3.37iT - 29T^{2} \) |
| 31 | \( 1 - 1.35iT - 31T^{2} \) |
| 41 | \( 1 + (-2.73 + 4.74i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 3.28iT - 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 + (-5.09 - 8.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.55 - 5.51i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.07 + 0.618i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.93 - 12.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.215 - 0.372i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.23T + 73T^{2} \) |
| 79 | \( 1 + (-6.26 - 3.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.83 - 11.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.75 - 5.63i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.69iT - 97T^{2} \) |
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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
−10.58235260147038968122355218207, −9.548196520867426574026029422053, −8.849677923712423129043227108760, −7.67523027647590936875549422870, −6.61512387105815908715031756291, −5.58650517377617925040079476387, −5.33294649048868622046110428388, −4.15123500333695926115613407553, −2.79403280024217220389614673279, −2.38541602757994576616269493022,
0.11118236243817074867446417338, 2.04698492328620201482504693639, 3.11599312474072769311880099811, 4.46073940919339191227599847734, 5.10459587901315978823383895821, 6.16046457932923171491741203378, 6.88965590649428919228289721063, 7.56059519988876059929894695159, 8.371894127440851737415979738464, 9.571395624690723323424739972205