| L(s) = 1 | + (1.73 − 2.06i)2-s + (1.51 + 8.60i)4-s + (14.6 − 40.1i)5-s + (−2.26 + 12.8i)7-s + (57.7 + 33.3i)8-s + (−57.5 − 99.7i)10-s + (−36.2 − 99.4i)11-s + (160. − 135. i)13-s + (22.5 + 26.9i)14-s + (37.4 − 13.6i)16-s + (229. − 132. i)17-s + (76.9 − 133. i)19-s + (367. + 64.8i)20-s + (−268. − 97.5i)22-s + (−538. + 94.8i)23-s + ⋯ |
| L(s) = 1 | + (0.433 − 0.516i)2-s + (0.0948 + 0.537i)4-s + (0.584 − 1.60i)5-s + (−0.0462 + 0.262i)7-s + (0.902 + 0.520i)8-s + (−0.575 − 0.997i)10-s + (−0.299 − 0.822i)11-s + (0.952 − 0.799i)13-s + (0.115 + 0.137i)14-s + (0.146 − 0.0532i)16-s + (0.793 − 0.458i)17-s + (0.213 − 0.369i)19-s + (0.919 + 0.162i)20-s + (−0.553 − 0.201i)22-s + (−1.01 + 0.179i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.95408 - 1.37766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95408 - 1.37766i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.73 + 2.06i)T + (-2.77 - 15.7i)T^{2} \) |
| 5 | \( 1 + (-14.6 + 40.1i)T + (-478. - 401. i)T^{2} \) |
| 7 | \( 1 + (2.26 - 12.8i)T + (-2.25e3 - 821. i)T^{2} \) |
| 11 | \( 1 + (36.2 + 99.4i)T + (-1.12e4 + 9.41e3i)T^{2} \) |
| 13 | \( 1 + (-160. + 135. i)T + (4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-229. + 132. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-76.9 + 133. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (538. - 94.8i)T + (2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-216. + 257. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-285. - 1.61e3i)T + (-8.67e5 + 3.15e5i)T^{2} \) |
| 37 | \( 1 + (-965. - 1.67e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-362. - 432. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (1.55e3 - 566. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (2.33e3 + 410. i)T + (4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 - 3.36e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (87.1 - 239. i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-201. + 1.14e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (1.29e3 - 1.09e3i)T + (3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-510. + 294. i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (767. - 1.32e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (6.87e3 + 5.76e3i)T + (6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (942. - 1.12e3i)T + (-8.24e6 - 4.67e7i)T^{2} \) |
| 89 | \( 1 + (3.80e3 + 2.19e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-5.83e3 + 2.12e3i)T + (6.78e7 - 5.69e7i)T^{2} \) |
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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
−13.29348060120063599176194696228, −12.43663136292148977797541139371, −11.57324417151914924058479175067, −10.14995897327916066996762114629, −8.716530359884588738212029241340, −8.032986953126837074536607292627, −5.83836538236763592379310180532, −4.76110604502341328609587283923, −3.13256165642197000610624455739, −1.21790696635514009882487901630,
1.98107366666396940241422629138, 3.93368864164768832849323988399, 5.79697349698958678148007634796, 6.58982984022398369474522883584, 7.62124785938769183354513107009, 9.793240862897782969978727863972, 10.38322632612704528757069718296, 11.45146073790491955835383681473, 13.19515531971514972109771383116, 14.16501969936057630113837354821
