| L(s) = 1 | + (−0.654 − 0.755i)2-s + (1.09 − 2.39i)3-s + (−0.142 + 0.989i)4-s + (0.959 + 0.281i)5-s + (−2.52 + 0.741i)6-s + (2.17 + 2.51i)7-s + (0.841 − 0.540i)8-s + (−2.57 − 2.97i)9-s + (−0.415 − 0.909i)10-s + (3.88 + 1.14i)11-s + (2.21 + 1.42i)12-s + (−1.38 − 0.886i)13-s + (0.473 − 3.29i)14-s + (1.72 − 1.99i)15-s + (−0.959 − 0.281i)16-s + (−0.0266 − 0.185i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.631 − 1.38i)3-s + (−0.0711 + 0.494i)4-s + (0.429 + 0.125i)5-s + (−1.03 + 0.302i)6-s + (0.823 + 0.950i)7-s + (0.297 − 0.191i)8-s + (−0.859 − 0.991i)9-s + (−0.131 − 0.287i)10-s + (1.17 + 0.344i)11-s + (0.639 + 0.411i)12-s + (−0.382 − 0.246i)13-s + (0.126 − 0.880i)14-s + (0.445 − 0.513i)15-s + (−0.239 − 0.0704i)16-s + (−0.00645 − 0.0448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30704 - 1.20314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30704 - 1.20314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (4.47 + 6.85i)T \) |
| good | 3 | \( 1 + (-1.09 + 2.39i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-2.17 - 2.51i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.88 - 1.14i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.886i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.0266 + 0.185i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 2.23i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.620 - 1.35i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 + (-5.86 + 3.76i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + (0.216 + 1.50i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.391 + 2.72i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (3.84 - 8.41i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-0.316 + 2.20i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (7.72 - 4.96i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (7.40 - 2.17i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (0.517 - 3.59i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.66 - 0.782i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (3.04 + 1.95i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (13.9 + 4.11i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-4.64 - 10.1i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.20003040862444694278490956481, −9.169731459144659274122856350513, −8.662613819962741760628138752001, −7.78259573332026259200707121240, −7.00466591162321347181141257246, −6.07231278138756407995439291538, −4.74755095427790274589386579662, −3.08046596132591012422297842628, −2.12947517911521079822175429939, −1.33653346772396017759231068591,
1.48398014021721751836917361922, 3.32854182296430790397328553632, 4.38135587691455983341537626934, 5.00132158174355519793246077048, 6.31768609953795725937686142573, 7.33544582562922981136862791437, 8.458149787060573886610091475330, 8.891330284407135840525452767489, 9.943478872233392258232121964282, 10.25477787162138353626599000549
