L(s) = 1 | + (3.66 + 3.68i)3-s + (−8.29 + 14.3i)5-s + (−6.28 + 17.4i)7-s + (−0.127 + 26.9i)9-s + (46.2 − 26.7i)11-s + (−11.0 − 6.37i)13-s + (−83.3 + 22.1i)15-s − 96.7·17-s − 54.6i·19-s + (−87.1 + 40.7i)21-s + (−55.6 − 32.1i)23-s + (−75.2 − 130. i)25-s + (−99.9 + 98.5i)27-s + (−112. + 65.0i)29-s + (190. + 110. i)31-s + ⋯ |
L(s) = 1 | + (0.705 + 0.708i)3-s + (−0.742 + 1.28i)5-s + (−0.339 + 0.940i)7-s + (−0.00470 + 0.999i)9-s + (1.26 − 0.732i)11-s + (−0.235 − 0.136i)13-s + (−1.43 + 0.380i)15-s − 1.38·17-s − 0.660i·19-s + (−0.906 + 0.423i)21-s + (−0.504 − 0.291i)23-s + (−0.601 − 1.04i)25-s + (−0.712 + 0.702i)27-s + (−0.721 + 0.416i)29-s + (1.10 + 0.637i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.424432837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424432837\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.66 - 3.68i)T \) |
| 7 | \( 1 + (6.28 - 17.4i)T \) |
good | 5 | \( 1 + (8.29 - 14.3i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-46.2 + 26.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (11.0 + 6.37i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (55.6 + 32.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (112. - 65.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-190. - 110. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (185. - 320. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (153. + 266. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-163. - 283. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 451. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-258. + 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-234. + 135. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (370. - 642. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 914. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 337. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (498. + 863. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-17.0 - 29.4i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.10e3 + 635. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.63579123577736223336104223357, −11.22134975055910173670610201787, −10.11415271064344491811588385133, −9.067368277200032756319144836875, −8.393379703790211331248314074882, −7.06728540534389974494801607582, −6.15905076696325018378537788008, −4.46677431190655788317040816577, −3.36826403145097574039839785676, −2.50400414996703955298423821527,
0.51206479762816035786765542357, 1.80110162713730102301654237897, 3.84230974143311907867352254755, 4.42633682828587348205565827597, 6.36451737248830121084696491669, 7.30100463601776284267300032227, 8.185531177910482329894512744646, 9.095099687311501401004473957434, 9.861361673195024553845920413053, 11.56273586736142487873998904340