L(s) = 1 | + (−0.951 + 1.64i)5-s + (1.46 + 2.20i)7-s + (−1.41 − 2.44i)11-s + (−2.41 − 4.17i)13-s + (−2.14 + 3.70i)17-s + (2.37 + 4.11i)19-s + (1.23 − 2.13i)23-s + (0.689 + 1.19i)25-s + (−4.32 + 7.49i)29-s + 3.37·31-s + (−5.02 + 0.310i)35-s + (−2.59 − 4.48i)37-s + (−4.10 − 7.10i)41-s + (−3.36 + 5.82i)43-s + 1.72·47-s + ⋯ |
L(s) = 1 | + (−0.425 + 0.737i)5-s + (0.552 + 0.833i)7-s + (−0.426 − 0.737i)11-s + (−0.669 − 1.15i)13-s + (−0.519 + 0.899i)17-s + (0.544 + 0.943i)19-s + (0.257 − 0.445i)23-s + (0.137 + 0.238i)25-s + (−0.803 + 1.39i)29-s + 0.605·31-s + (−0.849 + 0.0524i)35-s + (−0.426 − 0.738i)37-s + (−0.640 − 1.10i)41-s + (−0.513 + 0.888i)43-s + 0.252·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5949267620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5949267620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.46 - 2.20i)T \) |
good | 5 | \( 1 + (0.951 - 1.64i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.41 + 2.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.41 + 4.17i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.14 - 3.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 4.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 2.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.32 - 7.49i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + (2.59 + 4.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.10 + 7.10i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.36 - 5.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + (5.80 - 10.0i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 + (-4.23 + 7.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + (-4.72 + 8.18i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.91 + 8.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.60 - 2.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.223896763482208612540972740737, −8.545475674828977812739192419530, −7.81971890028618204498043272029, −7.27084401965529225204198085165, −6.10528771889800760855811255856, −5.54787903496025717680201823384, −4.69321501493903526594703198840, −3.38746310670134276045968328471, −2.89063358265434110390747384598, −1.64069603560517148522971356674,
0.19897664082314861196239603809, 1.54551301958280556558251149019, 2.65402556708463406779167782922, 4.00806872028867005346091128709, 4.80388823082813858619833600837, 4.98310304274452938582822181493, 6.56790920284625731123193625635, 7.20476174437376935502788227578, 7.80489665198030896664762506223, 8.632708741332828673348494744686