L(s) = 1 | + (−2.28 + 3.96i)2-s + (−6.45 − 11.1i)4-s + (2.5 + 4.33i)5-s + (10.0 − 17.4i)7-s + 22.4·8-s − 22.8·10-s + (33.1 − 57.4i)11-s + (23.4 + 40.5i)13-s + (45.9 + 79.6i)14-s + (0.237 − 0.411i)16-s + 47.6·17-s − 9.95·19-s + (32.2 − 55.9i)20-s + (151. + 262. i)22-s + (4.79 + 8.30i)23-s + ⋯ |
L(s) = 1 | + (−0.808 + 1.40i)2-s + (−0.807 − 1.39i)4-s + (0.223 + 0.387i)5-s + (0.543 − 0.940i)7-s + 0.993·8-s − 0.723·10-s + (0.909 − 1.57i)11-s + (0.499 + 0.864i)13-s + (0.878 + 1.52i)14-s + (0.00371 − 0.00643i)16-s + 0.679·17-s − 0.120·19-s + (0.361 − 0.625i)20-s + (1.47 + 2.54i)22-s + (0.0434 + 0.0753i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.01257 + 0.588571i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01257 + 0.588571i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (2.28 - 3.96i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-10.0 + 17.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-33.1 + 57.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-23.4 - 40.5i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 47.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.95T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-4.79 - 8.30i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (89.3 - 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-124. - 216. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-106. + 183. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-237. + 411. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (209. + 363. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-223. - 387. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 358.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-325. + 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (406. - 704. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (126. - 218. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.50792903439107953419524670005, −11.57622447919854887769206997700, −10.71005862134710269434846349672, −9.426001286079479951448172437398, −8.566803923803790250362758621263, −7.52146959219496767034201757852, −6.57032579213853760814771673729, −5.63782406773366193130652842118, −3.82378101477266373673975012897, −0.982715769674182569809337261981,
1.28117760485720698467757558440, 2.46606726067984353045417699574, 4.17291353941390207178393797955, 5.75205965696564852229317901696, 7.68194773378565401510079883432, 8.821984139012401475586786816176, 9.523437610370136138452870100445, 10.46952364991450494798308548235, 11.63400386630990849680060850315, 12.29049981315727158511076891642