L(s) = 1 | + (0.984 − 0.173i)2-s + (2.09 + 2.49i)3-s + (0.939 − 0.342i)4-s + (2.49 + 2.09i)6-s + (−0.964 − 0.556i)7-s + (0.866 − 0.5i)8-s + (−1.31 + 7.46i)9-s + (0.0761 + 0.131i)11-s + (2.81 + 1.62i)12-s + (−2.23 + 2.66i)13-s + (−1.04 − 0.380i)14-s + (0.766 − 0.642i)16-s + (0.164 − 0.0290i)17-s + 7.58i·18-s + (3.08 − 3.08i)19-s + ⋯ |
L(s) = 1 | + (0.696 − 0.122i)2-s + (1.20 + 1.43i)3-s + (0.469 − 0.171i)4-s + (1.01 + 0.853i)6-s + (−0.364 − 0.210i)7-s + (0.306 − 0.176i)8-s + (−0.438 + 2.48i)9-s + (0.0229 + 0.0397i)11-s + (0.813 + 0.469i)12-s + (−0.619 + 0.738i)13-s + (−0.279 − 0.101i)14-s + (0.191 − 0.160i)16-s + (0.0399 − 0.00704i)17-s + 1.78i·18-s + (0.707 − 0.706i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.53497 + 2.22147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.53497 + 2.22147i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.08 + 3.08i)T \) |
good | 3 | \( 1 + (-2.09 - 2.49i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.964 + 0.556i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0761 - 0.131i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.23 - 2.66i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.164 + 0.0290i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.12 - 5.83i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.881 - 4.99i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.34 + 7.52i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.71iT - 37T^{2} \) |
| 41 | \( 1 + (-3.70 + 3.10i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.46 + 9.50i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-8.28 - 1.46i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.87 + 13.3i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.35 + 13.3i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.92 - 1.79i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.55 + 0.450i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.70 - 0.622i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.53 - 7.79i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (4.00 - 3.35i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.26 + 3.03i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.18 + 7.70i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-12.2 + 2.16i)T + (91.1 - 33.1i)T^{2} \) |
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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
−9.986570463578471107855843345078, −9.559327216279030564945097472894, −8.836515236117540305681230407304, −7.73362742131097128660414858999, −6.94492656454487225480433224250, −5.46314591339375661634227374898, −4.74338325672724814963020019661, −3.86968454413483178716971466757, −3.15524277266027806234880600478, −2.16766402697596425136593033533,
1.18960397562885687475822385884, 2.65717137492988561687550312356, 3.00752956834814506747833407985, 4.35329699483095333070164248289, 5.81297574065222116787677998909, 6.46247111756481211548223323341, 7.45758025606781907911454683163, 7.86334274640162157760964971193, 8.806239566377280033238839597773, 9.627498515517130536367271950173