| L(s) = 1 | + (0.632 − 1.09i)5-s + (−13.4 − 12.7i)7-s + (36.0 − 20.7i)11-s − 85.7i·13-s + (−38.8 − 67.3i)17-s + (−42.1 − 24.3i)19-s + (78.7 + 45.4i)23-s + (61.6 + 106. i)25-s + 151. i·29-s + (−76.3 + 44.0i)31-s + (−22.4 + 6.60i)35-s + (−45.2 + 78.4i)37-s − 383.·41-s + 227.·43-s + (69.5 − 120. i)47-s + ⋯ |
| L(s) = 1 | + (0.0566 − 0.0980i)5-s + (−0.723 − 0.690i)7-s + (0.987 − 0.570i)11-s − 1.82i·13-s + (−0.554 − 0.961i)17-s + (−0.509 − 0.293i)19-s + (0.714 + 0.412i)23-s + (0.493 + 0.854i)25-s + 0.968i·29-s + (−0.442 + 0.255i)31-s + (−0.108 + 0.0318i)35-s + (−0.201 + 0.348i)37-s − 1.46·41-s + 0.808·43-s + (0.215 − 0.373i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8889763708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8889763708\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (13.4 + 12.7i)T \) |
| good | 5 | \( 1 + (-0.632 + 1.09i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-36.0 + 20.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 85.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (38.8 + 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (42.1 + 24.3i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-78.7 - 45.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 151. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (76.3 - 44.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (45.2 - 78.4i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-69.5 + 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-289. + 167. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (440. + 762. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 6.57i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (221. + 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 341. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (798. - 460. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (206. - 357. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-14.8 + 25.7i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
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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
−9.141512611327592694128206451434, −8.500342519474918789206780077290, −7.28799114668493951015052054663, −6.79326419294934410705220348633, −5.70857858817091301000630002101, −4.85093569973159004631958889095, −3.54601828344215048084023863044, −2.99292262770777617032276037297, −1.22052193089095717599548625978, −0.23049720313961390378609315516,
1.62089161398446707522444157410, 2.51494680967681846287709656601, 3.94639679367436815437723043304, 4.51233323569072903346989474082, 6.02993678770172618933274247245, 6.50170347561811876665474288497, 7.25790219130142472825773317264, 8.791545314461599187080254376014, 8.931392363439245394090144077956, 9.894788068819000773438489844853
