| L(s) = 1 | − 0.168i·2-s + (1.20 − 1.24i)3-s + 1.97·4-s + (−0.209 − 0.202i)6-s + 2.38·7-s − 0.668i·8-s + (−0.106 − 2.99i)9-s + 3.31i·11-s + (2.37 − 2.45i)12-s − 3.60·13-s − 0.400i·14-s + 3.83·16-s + (−0.504 + 0.0179i)18-s − 2.62·19-s + (2.86 − 2.96i)21-s + 0.557·22-s + ⋯ |
| L(s) = 1 | − 0.118i·2-s + (0.694 − 0.719i)3-s + 0.985·4-s + (−0.0855 − 0.0825i)6-s + 0.899·7-s − 0.236i·8-s + (−0.0355 − 0.999i)9-s + 1.00i·11-s + (0.684 − 0.709i)12-s − 1.00·13-s − 0.107i·14-s + 0.957·16-s + (−0.118 + 0.00423i)18-s − 0.601·19-s + (0.624 − 0.647i)21-s + 0.118·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02866 - 0.819236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02866 - 0.819236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.20 + 1.24i)T \) |
| 11 | \( 1 - 3.31iT \) |
| 13 | \( 1 + 3.60T \) |
| good | 2 | \( 1 + 0.168iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.62T + 19T^{2} \) |
| 23 | \( 1 - 5.04iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 12.1iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6.26iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.22286458215976693777299260343, −10.14213438568137146651073653942, −9.255676414852392032925521322506, −7.956002611658744938488650708396, −7.47778718485672955041622394344, −6.66719591792718251719263821151, −5.38414826991014095529200491776, −3.92431073606839139748766467014, −2.43427347727883874059389538905, −1.71081775562577338368494593443,
2.02099039855821395437794812419, 3.05406873123086287220757405890, 4.41308449523106049482691989213, 5.46353491424206323683701304257, 6.64872728146910323113488244955, 7.933521095655426505176020364820, 8.255123628061830325768919192831, 9.528695422127590091393160904096, 10.47352026602427446655068434695, 11.14722007055961940109123600678
