| L(s) = 1 | + (1.35 − 5.01i)3-s + (2.40 − 4.16i)5-s + (3.40 + 18.2i)7-s + (−23.3 − 13.5i)9-s + (−29.6 + 17.1i)11-s − 17.1i·13-s + (−17.6 − 17.6i)15-s + (−50.4 − 87.4i)17-s + (−119. − 69.2i)19-s + (95.9 + 7.51i)21-s + (−0.541 − 0.312i)23-s + (50.9 + 88.2i)25-s + (−99.5 + 98.8i)27-s − 222. i·29-s + (−247. + 143. i)31-s + ⋯ |
| L(s) = 1 | + (0.259 − 0.965i)3-s + (0.215 − 0.372i)5-s + (0.183 + 0.982i)7-s + (−0.864 − 0.501i)9-s + (−0.813 + 0.469i)11-s − 0.366i·13-s + (−0.303 − 0.304i)15-s + (−0.720 − 1.24i)17-s + (−1.44 − 0.835i)19-s + (0.996 + 0.0780i)21-s + (−0.00490 − 0.00283i)23-s + (0.407 + 0.705i)25-s + (−0.709 + 0.704i)27-s − 1.42i·29-s + (−1.43 + 0.829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5907968572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5907968572\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.35 + 5.01i)T \) |
| 7 | \( 1 + (-3.40 - 18.2i)T \) |
| good | 5 | \( 1 + (-2.40 + 4.16i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (29.6 - 17.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 17.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (50.4 + 87.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (119. + 69.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (0.541 + 0.312i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 222. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (247. - 143. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-9.19 + 15.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 370.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (143. - 247. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-29.3 + 16.9i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-127. - 220. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (29.2 + 16.8i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (87.9 + 152. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 626. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-189. + 109. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-263. + 456. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (412. - 714. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.30e3iT - 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83411456241040006253465954964, −9.338656862968032769023698021534, −8.781753074690969752053918923453, −7.78512238827853282355678160590, −6.83283191852449328507024134945, −5.72062725960724178011569257467, −4.77958956489563528839390017388, −2.80312817685566352143735370942, −1.98623941903398390348662875037, −0.18386036817522194869431753679,
2.10888159642603298704476291706, 3.59923256696362669411928066697, 4.38675207889924573488451183049, 5.66196852151355266398327363906, 6.77278760401778240873746479620, 8.085718645898707844681585716946, 8.756865620650571507555686689069, 10.03300898151456992577039330472, 10.73000263078801391221608786973, 11.02354850082561611038244686418
