L(s) = 1 | + (−1.64 − 0.546i)3-s + (2.41 − 1.08i)7-s + (2.40 + 1.79i)9-s + (−1.17 + 0.675i)11-s − 4.94·13-s + (4.97 − 2.87i)17-s + (−2.84 − 1.64i)19-s + (−4.55 + 0.460i)21-s + (2.50 − 4.33i)23-s + (−2.96 − 4.26i)27-s + 5.68i·29-s + (−2.45 + 1.41i)31-s + (2.29 − 0.471i)33-s + (−3.33 − 1.92i)37-s + (8.12 + 2.69i)39-s + ⋯ |
L(s) = 1 | + (−0.948 − 0.315i)3-s + (0.912 − 0.409i)7-s + (0.801 + 0.598i)9-s + (−0.353 + 0.203i)11-s − 1.37·13-s + (1.20 − 0.696i)17-s + (−0.653 − 0.377i)19-s + (−0.994 + 0.100i)21-s + (0.521 − 0.903i)23-s + (−0.571 − 0.820i)27-s + 1.05i·29-s + (−0.440 + 0.254i)31-s + (0.399 − 0.0820i)33-s + (−0.548 − 0.316i)37-s + (1.30 + 0.432i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8935467985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8935467985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.64 + 0.546i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.41 + 1.08i)T \) |
good | 11 | \( 1 + (1.17 - 0.675i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 + (-4.97 + 2.87i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.84 + 1.64i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.50 + 4.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.68iT - 29T^{2} \) |
| 31 | \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.33 + 1.92i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 4.06iT - 43T^{2} \) |
| 47 | \( 1 + (-4.92 - 2.84i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.730 - 1.26i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.34 + 7.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.65 - 0.954i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.36 + 2.51i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.38iT - 71T^{2} \) |
| 73 | \( 1 + (8.16 + 14.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.41 + 4.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.6iT - 83T^{2} \) |
| 89 | \( 1 + (8.08 - 13.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.831766065175755197051713358702, −7.73285023503289737224785510312, −7.34889289825172528389883353632, −6.63841934012747060666062599180, −5.38437281382004542325925709566, −5.03361694037685577111269576380, −4.24195201429831291072936594618, −2.75268993232918636068085515224, −1.65342652610385882856336754047, −0.38547566013158534955126691181,
1.25941679661512371591132922015, 2.46383221530904026233740840943, 3.81454663581099432565284068446, 4.66931213753414921602443329530, 5.48326549944369276924699862140, 5.86690121857109744666606242895, 7.09474909545809272670352095575, 7.73107874629690028797841909918, 8.520294403607588192101776605950, 9.624080848637578218641292031733