| L(s) = 1 | + (1.41 + i)3-s + 2.82i·5-s − 2i·7-s + (1.00 + 2.82i)9-s − 2.82·11-s + 2·13-s + (−2.82 + 4.00i)15-s − 6i·19-s + (2 − 2.82i)21-s + 5.65·23-s − 3.00·25-s + (−1.41 + 5.00i)27-s − 2.82i·29-s − 2i·31-s + (−4.00 − 2.82i)33-s + ⋯ |
| L(s) = 1 | + (0.816 + 0.577i)3-s + 1.26i·5-s − 0.755i·7-s + (0.333 + 0.942i)9-s − 0.852·11-s + 0.554·13-s + (−0.730 + 1.03i)15-s − 1.37i·19-s + (0.436 − 0.617i)21-s + 1.17·23-s − 0.600·25-s + (−0.272 + 0.962i)27-s − 0.525i·29-s − 0.359i·31-s + (−0.696 − 0.492i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30607 + 0.676072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30607 + 0.676072i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−13.13148637618050117541204373854, −11.22285971747827100843469849713, −10.71011108453119921109912175239, −9.889218324616254804192665148232, −8.698762734966275147682447725306, −7.53941410192935354153928842610, −6.75659233377462367178444720185, −4.99915589918558896935878875419, −3.60613380905763838820273751901, −2.60518331898954340030135478236,
1.55928948136384778523021942033, 3.21968927239413993215868846050, 4.87414875768836396992738561631, 6.02999253285561042268650256521, 7.54531832632720649279581234663, 8.552929065741834292129229860099, 8.971471352432460227181818175133, 10.23577994596983864776210178772, 11.76841780963787005279708874721, 12.68291779865034900842467329528
