| L(s) = 1 | + (−0.731 + 1.21i)2-s + (0.540 − 0.841i)3-s + (−0.929 − 1.77i)4-s + (3.52 + 1.60i)5-s + (0.622 + 1.26i)6-s + (4.53 + 1.33i)7-s + (2.82 + 0.169i)8-s + (−0.415 − 0.909i)9-s + (−4.52 + 3.08i)10-s + (−2.16 − 1.87i)11-s + (−1.99 − 0.175i)12-s + (1.08 + 3.70i)13-s + (−4.93 + 4.51i)14-s + (3.25 − 2.09i)15-s + (−2.27 + 3.29i)16-s + (−0.821 − 5.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.517 + 0.855i)2-s + (0.312 − 0.485i)3-s + (−0.464 − 0.885i)4-s + (1.57 + 0.719i)5-s + (0.254 + 0.518i)6-s + (1.71 + 0.503i)7-s + (0.998 + 0.0600i)8-s + (−0.138 − 0.303i)9-s + (−1.43 + 0.976i)10-s + (−0.651 − 0.564i)11-s + (−0.575 − 0.0505i)12-s + (0.301 + 1.02i)13-s + (−1.31 + 1.20i)14-s + (0.841 − 0.540i)15-s + (−0.567 + 0.823i)16-s + (−0.199 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59977 + 0.695969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59977 + 0.695969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.731 - 1.21i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (-0.624 + 4.75i)T \) |
| good | 5 | \( 1 + (-3.52 - 1.60i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-4.53 - 1.33i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.16 + 1.87i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.08 - 3.70i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.821 + 5.71i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (4.11 + 0.592i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (2.26 - 0.326i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.43 - 2.85i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.15 - 1.43i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (3.72 - 8.15i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.93 + 4.57i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 + (0.259 - 0.884i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (3.54 + 12.0i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 3.35i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.94 + 3.41i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-8.19 - 9.45i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.513 - 3.57i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (6.10 - 1.79i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.87 - 1.31i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (14.3 + 9.23i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.32 - 5.09i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.87917229928581033323686442817, −9.806355422697633209707124902062, −8.871406817364752914048307936215, −8.388640821658634781242673635391, −7.20386578682758463638593442276, −6.49765380653700848695581118079, −5.53594776173667178645482055150, −4.78844274307320447919262290810, −2.43472244962050897145506125281, −1.65033315441508560132736351721,
1.53471490900144078152086295076, 2.17218998865662908998803731686, 3.93090881404924217316056967810, 4.93010681541338525547794837839, 5.66472537691309273291197739338, 7.56784502213494098857041183758, 8.332362196670591137144287758825, 8.952892793791951811005812213433, 9.994783135398056687259510597615, 10.56040163727450545033723641424
