| L(s) = 1 | + i·3-s − 0.792i·5-s + 3.46·7-s − 9-s + 3.37i·11-s + 5.84i·13-s + 0.792·15-s − 17-s + 0.627i·19-s + 3.46i·21-s + 5.84·23-s + 4.37·25-s − i·27-s + 5.04i·29-s − 6.63·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.354i·5-s + 1.30·7-s − 0.333·9-s + 1.01i·11-s + 1.61i·13-s + 0.204·15-s − 0.242·17-s + 0.144i·19-s + 0.755i·21-s + 1.21·23-s + 0.874·25-s − 0.192i·27-s + 0.937i·29-s − 1.19·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927201028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927201028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 0.792iT - 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.37iT - 11T^{2} \) |
| 13 | \( 1 - 5.84iT - 13T^{2} \) |
| 19 | \( 1 - 0.627iT - 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 5.04iT - 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 + 3.16iT - 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 6.11iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 1.87iT - 53T^{2} \) |
| 59 | \( 1 - 2.74iT - 59T^{2} \) |
| 61 | \( 1 - 5.34iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.902307951213832504587792682785, −8.357202769973474900036256022176, −7.16820779226728103351489670074, −6.93224184408061289520490977152, −5.55673103050149638330476316775, −4.78035233151220408613878622874, −4.53699142426775996245585584455, −3.49883433206534173990013645525, −2.12873781690962140181878015216, −1.44227188342826731851423699486,
0.60242863589734796587190692136, 1.62873536493018476074239795153, 2.81693907621653321645051906960, 3.43240745386328841798426135708, 4.84161548867832528077434306728, 5.32128684851171332735802218526, 6.19754885479902958820350685662, 6.99962239825969591739159701443, 7.85613167089990326054598526043, 8.243642599403387595021653160397
