L(s) = 1 | + (−0.601 − 1.62i)3-s − 0.207·5-s + (−1.37 − 2.26i)7-s + (−2.27 + 1.95i)9-s − 1.51i·11-s + (−3.39 + 1.95i)13-s + (0.124 + 0.336i)15-s + (−0.873 − 1.51i)17-s + (−0.968 − 0.559i)19-s + (−2.85 + 3.58i)21-s + 1.44i·23-s − 4.95·25-s + (4.54 + 2.52i)27-s + (1.99 + 1.15i)29-s + (−5.15 − 2.97i)31-s + ⋯ |
L(s) = 1 | + (−0.347 − 0.937i)3-s − 0.0925·5-s + (−0.518 − 0.855i)7-s + (−0.759 + 0.651i)9-s − 0.457i·11-s + (−0.940 + 0.543i)13-s + (0.0321 + 0.0868i)15-s + (−0.211 − 0.366i)17-s + (−0.222 − 0.128i)19-s + (−0.622 + 0.782i)21-s + 0.301i·23-s − 0.991·25-s + (0.873 + 0.485i)27-s + (0.370 + 0.214i)29-s + (−0.926 − 0.534i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0223363 + 0.470154i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0223363 + 0.470154i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.601 + 1.62i)T \) |
| 7 | \( 1 + (1.37 + 2.26i)T \) |
good | 5 | \( 1 + 0.207T + 5T^{2} \) |
| 11 | \( 1 + 1.51iT - 11T^{2} \) |
| 13 | \( 1 + (3.39 - 1.95i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.873 + 1.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.968 + 0.559i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.44iT - 23T^{2} \) |
| 29 | \( 1 + (-1.99 - 1.15i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.15 + 2.97i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.13 - 3.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.91 + 5.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.213 - 0.369i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.13 + 7.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.16 + 4.71i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 - 6.07i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.24 + 5.62i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.10iT - 71T^{2} \) |
| 73 | \( 1 + (-10.6 + 6.13i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.57 + 4.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.36 + 14.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.96 + 3.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 + 7.57i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.55945219356715140989326658124, −9.638684516156021496651629680514, −8.546073429498125079782320973946, −7.45878315060275815205069872304, −6.95004383002640489671002905897, −5.96591629670206158336874335243, −4.81463256696644527984079985672, −3.46033574191847758495398468825, −2.01400516564623472298745799943, −0.27365169488911626955831402398,
2.45326713541236596590609789738, 3.65583158291698981930638432839, 4.82214059533911199644742175089, 5.66876521714010343261510619289, 6.60204719166888569022291774068, 7.902957572123216023765986415873, 8.943680165095461282911191399803, 9.684831989877218990743705626924, 10.35357987245725817289324491732, 11.31249738703780868073218371325