| L(s) = 1 | + (0.342 − 0.939i)2-s + (−1.71 − 0.222i)3-s + (−0.766 − 0.642i)4-s + (0.673 − 3.81i)5-s + (−0.796 + 1.53i)6-s + (−2.61 + 0.407i)7-s + (−0.866 + 0.500i)8-s + (2.90 + 0.763i)9-s + (−3.35 − 1.93i)10-s + (−1.75 + 0.309i)11-s + (1.17 + 1.27i)12-s + (0.186 + 0.512i)13-s + (−0.511 + 2.59i)14-s + (−2.00 + 6.41i)15-s + (0.173 + 0.984i)16-s + (0.365 − 0.633i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.991 − 0.128i)3-s + (−0.383 − 0.321i)4-s + (0.301 − 1.70i)5-s + (−0.325 + 0.627i)6-s + (−0.988 + 0.153i)7-s + (−0.306 + 0.176i)8-s + (0.967 + 0.254i)9-s + (−1.06 − 0.613i)10-s + (−0.529 + 0.0932i)11-s + (0.338 + 0.367i)12-s + (0.0517 + 0.142i)13-s + (−0.136 + 0.693i)14-s + (−0.517 + 1.65i)15-s + (0.0434 + 0.246i)16-s + (0.0886 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.140444 + 0.525244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140444 + 0.525244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 3 | \( 1 + (1.71 + 0.222i)T \) |
| 7 | \( 1 + (2.61 - 0.407i)T \) |
| good | 5 | \( 1 + (-0.673 + 3.81i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.75 - 0.309i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.186 - 0.512i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.633i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.82 - 2.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.91 + 2.28i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.179 - 0.493i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.55 - 3.04i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.56 + 4.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.29 - 3.02i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.17 + 12.3i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.84 + 3.23i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 6.08iT - 53T^{2} \) |
| 59 | \( 1 + (-2.16 + 12.2i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.59 + 6.66i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.60 + 3.49i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 5.13i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.00 + 1.73i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.95 + 3.62i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.14 - 2.23i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.0632 - 0.109i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 1.22i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.83901516677564076693852162903, −10.02994350487731864873139619170, −9.217229625517204375329339088895, −8.291160392296559144106507535671, −6.71887961050523205056537321030, −5.64677012967534682889030301194, −4.98743929511474343578004432332, −3.91028390253192645574576940137, −1.91222687283148408607216033347, −0.36702910582174689490273924433,
2.79737636287061872261117136035, 3.96199007962987325022643358621, 5.43167739000117559826775550531, 6.41162927173168578210745243747, 6.74831893637226054975197916054, 7.74161042190783112584074347484, 9.395470116184234063612942072281, 10.26115280018261390226087022127, 10.83455980442199783752789226686, 11.76573292808503183412324564782
