| L(s) = 1 | + (−12.9 − 4.71i)2-s + (−37.0 + 135. i)3-s + (−246. − 206. i)4-s + (−222. + 1.26e3i)5-s + (1.11e3 − 1.57e3i)6-s + (−4.99e3 + 4.19e3i)7-s + (5.74e3 + 9.95e3i)8-s + (−1.69e4 − 1.00e4i)9-s + (8.83e3 − 1.52e4i)10-s + (4.38e3 + 2.48e4i)11-s + (3.71e4 − 2.57e4i)12-s + (1.45e4 − 5.29e3i)13-s + (8.45e4 − 3.07e4i)14-s + (−1.62e5 − 7.68e4i)15-s + (1.08e3 + 6.16e3i)16-s + (1.87e5 − 3.24e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.208i)2-s + (−0.263 + 0.964i)3-s + (−0.481 − 0.404i)4-s + (−0.159 + 0.902i)5-s + (0.352 − 0.497i)6-s + (−0.786 + 0.660i)7-s + (0.496 + 0.859i)8-s + (−0.860 − 0.509i)9-s + (0.279 − 0.483i)10-s + (0.0903 + 0.512i)11-s + (0.516 − 0.357i)12-s + (0.141 − 0.0514i)13-s + (0.588 − 0.214i)14-s + (−0.828 − 0.391i)15-s + (0.00414 + 0.0235i)16-s + (0.543 − 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0449 + 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0449 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.100605 - 0.105232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100605 - 0.105232i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (37.0 - 135. i)T \) |
| good | 2 | \( 1 + (12.9 + 4.71i)T + (392. + 329. i)T^{2} \) |
| 5 | \( 1 + (222. - 1.26e3i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (4.99e3 - 4.19e3i)T + (7.00e6 - 3.97e7i)T^{2} \) |
| 11 | \( 1 + (-4.38e3 - 2.48e4i)T + (-2.21e9 + 8.06e8i)T^{2} \) |
| 13 | \( 1 + (-1.45e4 + 5.29e3i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (-1.87e5 + 3.24e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (3.31e5 + 5.74e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.82e6 + 1.52e6i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-2.27e6 - 8.28e5i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-4.94e6 - 4.14e6i)T + (4.59e12 + 2.60e13i)T^{2} \) |
| 37 | \( 1 + (8.89e6 - 1.54e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-3.98e6 + 1.45e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (2.25e6 + 1.27e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (1.91e6 - 1.60e6i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + 9.09e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-2.51e7 + 1.42e8i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (1.76e6 - 1.48e6i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (2.75e8 - 1.00e8i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-3.55e7 + 6.15e7i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (2.15e8 + 3.73e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.19e8 - 1.16e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (1.63e8 + 5.94e7i)T + (1.43e17 + 1.20e17i)T^{2} \) |
| 89 | \( 1 + (2.35e8 + 4.08e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.26e8 - 7.18e8i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−15.06085530319081550120007275381, −14.01712380513859570385392932066, −12.01439560120024557884872484162, −10.61082356842907483937968765174, −9.835981739788329826078998666887, −8.678749689625614000757082398103, −6.42759789635953990283142330905, −4.77882970066942847223890131060, −2.85593822366668823129246005216, −0.090984729357914088159641309584,
1.12851909437573099871880758947, 3.88700804334107272368299444683, 6.06303240625046720255567629936, 7.68786964915040084644813181938, 8.589102850100454463293702036115, 10.14908010603889216330629218138, 12.12372653581014063551305093891, 12.98207890715725587461462265357, 13.89935848234108756935402460669, 16.24390919428085548454098539818
