| 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 | \( 1 + (-3.06e3 + 1.76e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.90004300105861430516326248539, −14.10070698783429260506093926003, −13.34646711935797399904169424763, −11.41799425117633785811635066801, −10.58251182430041846130212049361, −8.294799337769516230704719150123, −7.33693109014896126837421940492, −6.37689163089038520024120337835, −4.30296510296276389066680010205, −1.86389326575647883747962913627,
2.22367611483434467847563879423, 4.04410826024384728952334120961, 5.15045654841974164213751673329, 8.275674870073897909119417535589, 8.495974779018805089708610746419, 10.66815769780675068985888402125, 11.38643951845415859194232952539, 12.73231250459558068176085816083, 13.69559247186352502360939647444, 15.19954053389929479437770295537
