| L(s) = 1 | + (−1.25 − 0.656i)2-s + (2.52 − 0.633i)3-s + (1.13 + 1.64i)4-s + (2.68 + 1.27i)5-s + (−3.58 − 0.865i)6-s + (−0.994 − 3.27i)7-s + (−0.347 − 2.80i)8-s + (3.34 − 1.78i)9-s + (−2.53 − 3.35i)10-s + (−2.32 + 0.345i)11-s + (3.91 + 3.43i)12-s + (−1.83 + 5.11i)13-s + (−0.905 + 4.76i)14-s + (7.59 + 1.51i)15-s + (−1.40 + 3.74i)16-s + (4.84 − 0.964i)17-s + ⋯ |
| L(s) = 1 | + (−0.885 − 0.464i)2-s + (1.45 − 0.365i)3-s + (0.569 + 0.822i)4-s + (1.20 + 0.568i)5-s + (−1.46 − 0.353i)6-s + (−0.376 − 1.23i)7-s + (−0.122 − 0.992i)8-s + (1.11 − 0.595i)9-s + (−0.800 − 1.06i)10-s + (−0.701 + 0.104i)11-s + (1.13 + 0.991i)12-s + (−0.507 + 1.41i)13-s + (−0.242 + 1.27i)14-s + (1.96 + 0.390i)15-s + (−0.351 + 0.936i)16-s + (1.17 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37801 - 0.498453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37801 - 0.498453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 + 0.656i)T \) |
| good | 3 | \( 1 + (-2.52 + 0.633i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-2.68 - 1.27i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.994 + 3.27i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (2.32 - 0.345i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (1.83 - 5.11i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-4.84 + 0.964i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (0.606 + 0.550i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (6.50 - 0.640i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (5.25 + 7.09i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-3.83 - 1.58i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.468 - 9.53i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-5.46 + 6.66i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-0.150 + 0.601i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (0.817 + 0.546i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (4.03 - 5.43i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (1.66 + 4.65i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (4.79 - 8.00i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-2.67 - 1.60i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-5.49 + 10.2i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (0.366 - 1.20i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (1.87 + 2.79i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.420 - 8.56i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-3.56 - 0.351i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-4.65 + 11.2i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.86478591437422194150022297777, −10.43984920221495877990335619617, −9.849310328237540453945633364331, −9.320389673600111093451024619084, −7.958835390242362677347355284872, −7.35384851737440746021799284015, −6.36759746653489665023659528897, −3.95871285941838909669979009396, −2.73715691038697993091306724687, −1.80878266500867574535134541085,
2.04673420675587408113144151234, 2.97271230093699290745706760521, 5.37318890583290956142550245789, 5.89331479271918286347339079076, 7.77810840653044535849571796615, 8.310595867772531098011515768901, 9.336826744819038628964624698003, 9.705553612906536238333200390288, 10.53513452899770888182786564417, 12.37755308129528085447831142644
