L(s) = 1 | + (0.388 − 3.45i)2-s + (−7.86 − 1.79i)4-s + (−4.44 + 3.54i)5-s + (1.25 + 5.49i)7-s + (−4.66 + 13.3i)8-s + (10.5 + 16.7i)10-s + (6.29 + 17.9i)11-s + (6.68 − 13.8i)13-s + (19.4 − 2.19i)14-s + (15.1 + 7.28i)16-s + (−1.08 − 1.08i)17-s + (−17.6 + 11.1i)19-s + (41.3 − 19.9i)20-s + (64.5 − 14.7i)22-s + (−4.81 + 6.03i)23-s + ⋯ |
L(s) = 1 | + (0.194 − 1.72i)2-s + (−1.96 − 0.448i)4-s + (−0.889 + 0.709i)5-s + (0.179 + 0.784i)7-s + (−0.583 + 1.66i)8-s + (1.05 + 1.67i)10-s + (0.572 + 1.63i)11-s + (0.514 − 1.06i)13-s + (1.38 − 0.156i)14-s + (0.946 + 0.455i)16-s + (−0.0636 − 0.0636i)17-s + (−0.930 + 0.584i)19-s + (2.06 − 0.995i)20-s + (2.93 − 0.669i)22-s + (−0.209 + 0.262i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00200i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.979455 + 0.000983720i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.979455 + 0.000983720i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 + (-18.9 - 21.9i)T \) |
good | 2 | \( 1 + (-0.388 + 3.45i)T + (-3.89 - 0.890i)T^{2} \) |
| 5 | \( 1 + (4.44 - 3.54i)T + (5.56 - 24.3i)T^{2} \) |
| 7 | \( 1 + (-1.25 - 5.49i)T + (-44.1 + 21.2i)T^{2} \) |
| 11 | \( 1 + (-6.29 - 17.9i)T + (-94.6 + 75.4i)T^{2} \) |
| 13 | \( 1 + (-6.68 + 13.8i)T + (-105. - 132. i)T^{2} \) |
| 17 | \( 1 + (1.08 + 1.08i)T + 289iT^{2} \) |
| 19 | \( 1 + (17.6 - 11.1i)T + (156. - 325. i)T^{2} \) |
| 23 | \( 1 + (4.81 - 6.03i)T + (-117. - 515. i)T^{2} \) |
| 31 | \( 1 + (5.79 - 51.4i)T + (-936. - 213. i)T^{2} \) |
| 37 | \( 1 + (0.580 - 1.65i)T + (-1.07e3 - 853. i)T^{2} \) |
| 41 | \( 1 + (0.198 - 0.198i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (16.5 - 1.86i)T + (1.80e3 - 411. i)T^{2} \) |
| 47 | \( 1 + (-2.91 + 1.02i)T + (1.72e3 - 1.37e3i)T^{2} \) |
| 53 | \( 1 + (31.3 + 39.3i)T + (-625. + 2.73e3i)T^{2} \) |
| 59 | \( 1 - 1.15T + 3.48e3T^{2} \) |
| 61 | \( 1 + (12.3 - 19.7i)T + (-1.61e3 - 3.35e3i)T^{2} \) |
| 67 | \( 1 + (-2.18 - 4.54i)T + (-2.79e3 + 3.50e3i)T^{2} \) |
| 71 | \( 1 + (12.0 - 24.9i)T + (-3.14e3 - 3.94e3i)T^{2} \) |
| 73 | \( 1 + (8.52 + 75.7i)T + (-5.19e3 + 1.18e3i)T^{2} \) |
| 79 | \( 1 + (-27.2 - 9.54i)T + (4.87e3 + 3.89e3i)T^{2} \) |
| 83 | \( 1 + (-5.13 + 22.4i)T + (-6.20e3 - 2.98e3i)T^{2} \) |
| 89 | \( 1 + (8.12 - 72.1i)T + (-7.72e3 - 1.76e3i)T^{2} \) |
| 97 | \( 1 + (-65.8 - 104. i)T + (-4.08e3 + 8.47e3i)T^{2} \) |
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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
−11.95598439684649651013042896719, −10.81324104800035640099743974667, −10.29924776905207479321368791676, −9.181179348180615895027421300734, −8.158404699513918533891588398699, −6.81700586740563004902744471861, −5.11831309145381657272251139257, −3.98277613267186956459295307686, −3.01076722161605189123580398169, −1.70870579198596167704256045418,
0.49738664928195897046771968995, 3.98385175891948523851995500794, 4.44188710559907311739460422712, 5.95545802851258856102592394957, 6.69733669854987350848782386566, 7.87631028367382885170219231711, 8.483040319904557328590031299042, 9.215540609327090537079371892664, 10.95943892545005861466463241584, 11.79359916012074475404642220010