| L(s) = 1 | + (1.64 − 2.85i)2-s + (5.28 + 7.28i)3-s + (2.56 + 4.44i)4-s + (−9.73 − 23.0i)5-s + (29.5 − 3.07i)6-s + (64.7 + 37.3i)7-s + 69.6·8-s + (−25.1 + 76.9i)9-s + (−81.7 − 10.1i)10-s + (−90.0 − 51.9i)11-s + (−18.8 + 42.2i)12-s + (183. − 105. i)13-s + (213. − 123. i)14-s + (116. − 192. i)15-s + (73.7 − 127. i)16-s − 321.·17-s + ⋯ |
| L(s) = 1 | + (0.412 − 0.713i)2-s + (0.587 + 0.809i)3-s + (0.160 + 0.277i)4-s + (−0.389 − 0.921i)5-s + (0.819 − 0.0855i)6-s + (1.32 + 0.762i)7-s + 1.08·8-s + (−0.310 + 0.950i)9-s + (−0.817 − 0.101i)10-s + (−0.744 − 0.429i)11-s + (−0.130 + 0.293i)12-s + (1.08 − 0.626i)13-s + (1.08 − 0.628i)14-s + (0.516 − 0.856i)15-s + (0.288 − 0.498i)16-s − 1.11·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0900i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.37238 - 0.107010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37238 - 0.107010i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.28 - 7.28i)T \) |
| 5 | \( 1 + (9.73 + 23.0i)T \) |
| good | 2 | \( 1 + (-1.64 + 2.85i)T + (-8 - 13.8i)T^{2} \) |
| 7 | \( 1 + (-64.7 - 37.3i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (90.0 + 51.9i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-183. + 105. i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 321.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 422.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (220. + 382. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-599. - 346. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (381. + 660. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 158. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-132. + 76.6i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.83e3 + 1.05e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.05e3 + 1.83e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 1.65e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (2.89e3 - 1.67e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.04e3 + 3.54e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-6.15e3 + 3.55e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 4.65e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.91e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.36e3 - 2.36e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (2.03e3 - 3.52e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 9.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.06e3 - 1.76e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
| 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
−15.19954053389929479437770295537, −13.69559247186352502360939647444, −12.73231250459558068176085816083, −11.38643951845415859194232952539, −10.66815769780675068985888402125, −8.495974779018805089708610746419, −8.275674870073897909119417535589, −5.15045654841974164213751673329, −4.04410826024384728952334120961, −2.22367611483434467847563879423,
1.86389326575647883747962913627, 4.30296510296276389066680010205, 6.37689163089038520024120337835, 7.33693109014896126837421940492, 8.294799337769516230704719150123, 10.58251182430041846130212049361, 11.41799425117633785811635066801, 13.34646711935797399904169424763, 14.10070698783429260506093926003, 14.90004300105861430516326248539
