| L(s) = 1 | + (1.36 − 0.363i)2-s + (0.0531 − 0.0267i)3-s + (1.73 − 0.993i)4-s + (0.468 + 1.05i)5-s + (0.0629 − 0.0558i)6-s + (1.73 + 1.04i)7-s + (2.01 − 1.98i)8-s + (−1.78 + 2.40i)9-s + (1.02 + 1.27i)10-s + (1.58 + 1.23i)11-s + (0.0656 − 0.0992i)12-s + (0.0910 − 0.0866i)13-s + (2.75 + 0.792i)14-s + (0.0531 + 0.0436i)15-s + (2.02 − 3.44i)16-s + (−0.562 − 0.685i)17-s + ⋯ |
| L(s) = 1 | + (0.966 − 0.256i)2-s + (0.0306 − 0.0154i)3-s + (0.867 − 0.496i)4-s + (0.209 + 0.472i)5-s + (0.0256 − 0.0228i)6-s + (0.656 + 0.393i)7-s + (0.711 − 0.702i)8-s + (−0.594 + 0.802i)9-s + (0.323 + 0.402i)10-s + (0.477 + 0.372i)11-s + (0.0189 − 0.0286i)12-s + (0.0252 − 0.0240i)13-s + (0.736 + 0.211i)14-s + (0.0137 + 0.0112i)15-s + (0.506 − 0.862i)16-s + (−0.136 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.78438 - 0.0416525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.78438 - 0.0416525i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.363i)T \) |
| good | 3 | \( 1 + (-0.0531 + 0.0267i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.468 - 1.05i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-1.73 - 1.04i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 1.23i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.0910 + 0.0866i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.562 + 0.685i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.292 + 1.68i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (4.49 + 2.12i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-0.798 - 0.452i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-0.612 + 0.121i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (7.59 + 4.80i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (6.35 + 7.01i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.58 - 7.82i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-11.2 - 5.99i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (4.04 - 2.29i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (4.55 - 4.78i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.0972 - 1.31i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (9.06 + 7.82i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (3.89 + 0.578i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-3.52 + 2.11i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-8.65 - 2.62i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (5.63 - 3.57i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (4.86 - 2.29i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (3.29 + 2.19i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.87768453680070988291491657978, −10.48732621437405215641897935024, −9.180222413882715104993104804822, −8.070265310446793608444520207072, −7.05430865419673553547850713327, −6.08662963337493321439870498731, −5.16626472356058037805112378007, −4.25804908976926983082208449343, −2.81909012081928411662103551564, −1.92179936505412723723403650578,
1.58989263531492226868748907462, 3.25668384020374486930177983093, 4.18989923880712513190577532854, 5.28155155853258783323785877431, 6.11218928816185673645318050524, 7.06127801809972415134522133486, 8.190351914871946488000007704307, 8.883175419631574012871989074597, 10.17736668400935626431533461791, 11.20369056374467750064649232833
