| L(s) = 1 | + (−0.378 + 1.36i)2-s + (0.342 − 0.939i)3-s + (−1.71 − 1.03i)4-s + (−2.71 + 0.479i)5-s + (1.15 + 0.821i)6-s + (−1.39 + 2.41i)7-s + (2.05 − 1.94i)8-s + (−0.766 − 0.642i)9-s + (0.374 − 3.88i)10-s + (5.27 − 3.04i)11-s + (−1.55 + 1.25i)12-s + (−0.786 − 2.16i)13-s + (−2.76 − 2.81i)14-s + (−0.479 + 2.71i)15-s + (1.87 + 3.53i)16-s + (2.16 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.267 + 0.963i)2-s + (0.197 − 0.542i)3-s + (−0.857 − 0.515i)4-s + (−1.21 + 0.214i)5-s + (0.470 + 0.335i)6-s + (−0.527 + 0.913i)7-s + (0.725 − 0.688i)8-s + (−0.255 − 0.214i)9-s + (0.118 − 1.22i)10-s + (1.58 − 0.917i)11-s + (−0.448 + 0.363i)12-s + (−0.218 − 0.599i)13-s + (−0.739 − 0.752i)14-s + (−0.123 + 0.702i)15-s + (0.469 + 0.883i)16-s + (0.524 − 0.440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761647 - 0.223073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761647 - 0.223073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.378 - 1.36i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (0.359 + 4.34i)T \) |
| good | 5 | \( 1 + (2.71 - 0.479i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.39 - 2.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.27 + 3.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.786 + 2.16i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 1.81i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.07 + 6.12i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.49 + 5.36i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.36 - 4.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.19iT - 37T^{2} \) |
| 41 | \( 1 + (-0.915 - 0.333i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (9.76 - 1.72i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.90 + 1.60i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-2.92 - 0.515i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.93 - 5.88i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.77 - 1.01i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.91 - 2.27i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.854 - 4.84i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.15 - 2.24i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 0.638i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (12.2 + 7.09i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.58 - 0.940i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.53 - 1.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.12216605708185423112569060451, −9.756113233185754325167685761484, −8.730367967370080816763198492058, −8.394957246834586777881651763732, −7.19484057635779686267411260283, −6.57454108274654478534844612499, −5.59943088849137601297175579225, −4.18239214258636285956272804122, −3.03052891053277314700316957267, −0.59678080801509639616826906400,
1.45187837096624758662018874155, 3.64748084717184412409238047673, 3.79776103643004654527934466591, 4.84239105778567853020626696263, 6.78513245869513335752776370187, 7.72039361435478364845032491531, 8.629888323624029273155924225550, 9.665913512673897937713607217151, 10.07223273678704028358491347975, 11.27149534549964646893027568939
