L(s) = 1 | + (−1.40 − 0.167i)2-s + (1.27 + 0.529i)3-s + (1.94 + 0.470i)4-s + (−0.707 − 1.70i)5-s + (−1.70 − 0.957i)6-s + (−2.74 + 2.74i)7-s + (−2.65 − 0.985i)8-s + (−0.766 − 0.766i)9-s + (0.707 + 2.51i)10-s + (0.135 − 0.0560i)11-s + (2.23 + 1.63i)12-s + (1.18 − 2.85i)13-s + (4.32 − 3.40i)14-s − 2.55i·15-s + (3.55 + 1.82i)16-s + 6.44i·17-s + ⋯ |
L(s) = 1 | + (−0.992 − 0.118i)2-s + (0.738 + 0.305i)3-s + (0.971 + 0.235i)4-s + (−0.316 − 0.763i)5-s + (−0.696 − 0.391i)6-s + (−1.03 + 1.03i)7-s + (−0.937 − 0.348i)8-s + (−0.255 − 0.255i)9-s + (0.223 + 0.795i)10-s + (0.0408 − 0.0169i)11-s + (0.645 + 0.470i)12-s + (0.327 − 0.790i)13-s + (1.15 − 0.908i)14-s − 0.660i·15-s + (0.889 + 0.457i)16-s + 1.56i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.545990 - 0.00119456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.545990 - 0.00119456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.167i)T \) |
good | 3 | \( 1 + (-1.27 - 0.529i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.707 + 1.70i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (2.74 - 2.74i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.135 + 0.0560i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.18 + 2.85i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 6.44iT - 17T^{2} \) |
| 19 | \( 1 + (-0.805 + 1.94i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.749 - 0.749i)T + 23iT^{2} \) |
| 29 | \( 1 + (-4.32 - 1.79i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + (-1.73 - 4.18i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (2.49 + 2.49i)T + 41iT^{2} \) |
| 43 | \( 1 + (6.10 - 2.52i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 2.66iT - 47T^{2} \) |
| 53 | \( 1 + (-1.64 + 0.682i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.43 - 3.47i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (3.46 + 1.43i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (14.0 + 5.83i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (3.40 - 3.40i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.442 + 0.442i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.07iT - 79T^{2} \) |
| 83 | \( 1 + (-2.99 + 7.23i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.21 - 4.21i)T - 89iT^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−16.84670491316641671520896865802, −15.68551673670633767509785236264, −15.08321500145372671801757480710, −12.93191992295355812935202824149, −11.96726639454369429514390160547, −10.17521481345542911491279484940, −8.930149116225319517445708874826, −8.340579096666816766094023682827, −6.16220400041491996203972656184, −3.14701352631460913060268324442,
3.06014740097946193535272357069, 6.71070073792761251293521555067, 7.57790062811542124808616291238, 9.140540461896382787966412353491, 10.36775388008133967301600583236, 11.60786617859072439796066293445, 13.53730427185499292584900409897, 14.50867896782748005174705761216, 15.99356456208478793607533818136, 16.78822038716304393768201487142