| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.600 + 1.62i)3-s + (−0.939 + 0.342i)4-s + (−1.35 + 3.71i)5-s + (−1.49 + 0.873i)6-s + (0.939 + 2.47i)7-s + (−0.5 − 0.866i)8-s + (−2.27 + 1.95i)9-s + (−3.88 − 0.685i)10-s − 2.71i·11-s + (−1.12 − 1.32i)12-s + (3.46 + 0.610i)13-s + (−2.27 + 1.35i)14-s + (−6.84 + 0.0351i)15-s + (0.766 − 0.642i)16-s + (0.0364 + 0.0434i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.346 + 0.937i)3-s + (−0.469 + 0.171i)4-s + (−0.604 + 1.65i)5-s + (−0.610 + 0.356i)6-s + (0.355 + 0.934i)7-s + (−0.176 − 0.306i)8-s + (−0.759 + 0.650i)9-s + (−1.22 − 0.216i)10-s − 0.818i·11-s + (−0.323 − 0.381i)12-s + (0.960 + 0.169i)13-s + (−0.607 + 0.362i)14-s + (−1.76 + 0.00907i)15-s + (0.191 − 0.160i)16-s + (0.00884 + 0.0105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.392735 - 1.49226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392735 - 1.49226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.600 - 1.62i)T \) |
| 7 | \( 1 + (-0.939 - 2.47i)T \) |
| 19 | \( 1 + (-1.41 - 4.12i)T \) |
| good | 5 | \( 1 + (1.35 - 3.71i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + (-3.46 - 0.610i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0364 - 0.0434i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-2.76 - 0.487i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.757 - 0.275i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-8.15 + 4.70i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.82 + 3.36i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.280 + 1.59i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.84 + 5.74i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.85 + 3.39i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (8.67 - 3.15i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (5.03 - 4.22i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0164 - 0.0934i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (11.8 + 2.08i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (9.86 - 8.27i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-8.25 - 3.00i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (0.766 + 0.913i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.35 - 4.24i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.32 - 1.93i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.03 - 8.34i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.85722199323111149896715962647, −9.883057607940081562703082948202, −8.904426788241451614969861346586, −8.181107233140982294586702786000, −7.47359894412252973183161393589, −6.15862621275978406988015971522, −5.75934160222991590025277941329, −4.30930441697688381898400220928, −3.45007369964738209143054928679, −2.65411847540649230402113452434,
0.800704719580997226127077974898, 1.46980598779442705659199197468, 3.11506982641617732370971872529, 4.36555263509145090319273640811, 4.88325870447008983981821079916, 6.28996527462465235463976620272, 7.50697384173217384199994634800, 8.107972595907497308016194701757, 8.893731207261697481192198263226, 9.581612419827016419847185902169
