| L(s) = 1 | + (0.611 + 1.90i)2-s + (−0.704 + 2.91i)3-s + (−3.25 + 2.32i)4-s + (−1.59 − 9.02i)5-s + (−5.98 + 0.441i)6-s + (−0.928 − 0.778i)7-s + (−6.42 − 4.77i)8-s + (−8.00 − 4.10i)9-s + (16.2 − 8.54i)10-s + (3.17 − 17.9i)11-s + (−4.49 − 11.1i)12-s + (−4.20 + 11.5i)13-s + (0.915 − 2.24i)14-s + (27.4 + 1.71i)15-s + (5.16 − 15.1i)16-s + (−16.5 + 9.56i)17-s + ⋯ |
| L(s) = 1 | + (0.305 + 0.952i)2-s + (−0.234 + 0.972i)3-s + (−0.813 + 0.582i)4-s + (−0.318 − 1.80i)5-s + (−0.997 + 0.0736i)6-s + (−0.132 − 0.111i)7-s + (−0.802 − 0.596i)8-s + (−0.889 − 0.456i)9-s + (1.62 − 0.854i)10-s + (0.288 − 1.63i)11-s + (−0.374 − 0.927i)12-s + (−0.323 + 0.889i)13-s + (0.0654 − 0.160i)14-s + (1.82 + 0.114i)15-s + (0.322 − 0.946i)16-s + (−0.974 + 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.496713 - 0.290625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496713 - 0.290625i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.611 - 1.90i)T \) |
| 3 | \( 1 + (0.704 - 2.91i)T \) |
| good | 5 | \( 1 + (1.59 + 9.02i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (0.928 + 0.778i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-3.17 + 17.9i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.20 - 11.5i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (16.5 - 9.56i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (21.0 + 12.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-15.5 - 18.5i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (3.71 - 1.35i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-17.2 + 14.4i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-5.34 + 3.08i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-10.7 + 29.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (55.2 + 9.74i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-10.6 + 12.7i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 55.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-5.56 - 31.5i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-13.9 + 16.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (11.1 - 30.6i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-62.2 + 35.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (19.3 - 33.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-65.6 + 23.9i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-25.2 + 9.17i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (104. + 60.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (31.1 - 176. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.98538941432908619185684043988, −11.14266581123338794743530345311, −9.460318067663799358263680355392, −8.798301188708457623889206624650, −8.333827663364990910132800504889, −6.51719217153809734874475776582, −5.45897470077020230691796049358, −4.55340113980478907667566043372, −3.77018199180527414719094681264, −0.28683104786462434218931310948,
2.13541153785511230659991747927, 2.98828866925841126716686446393, 4.62681303696968437690474197432, 6.29397932259969896118779809070, 6.97713411418952757796974703813, 8.141696294220818477703947733651, 9.734933159317166993323312791222, 10.63127601708118036799595667812, 11.28175091418993309577132312388, 12.29502528953180897974465974305
