| L(s) = 1 | + 2.23·5-s + 10.1i·7-s − 14.4i·11-s − 11.5·13-s + 18.9·17-s − 12i·19-s − 17.5i·23-s + 5.00·25-s + 8.83·29-s − 0.583i·31-s + 22.7i·35-s − 32.4·37-s − 71.3·41-s + 4.65i·43-s + 22.5i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.45i·7-s − 1.31i·11-s − 0.886·13-s + 1.11·17-s − 0.631i·19-s − 0.765i·23-s + 0.200·25-s + 0.304·29-s − 0.0188i·31-s + 0.650i·35-s − 0.877·37-s − 1.73·41-s + 0.108i·43-s + 0.479i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.376507265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376507265\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 - 10.1iT - 49T^{2} \) |
| 11 | \( 1 + 14.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.5T + 169T^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 19 | \( 1 + 12iT - 361T^{2} \) |
| 23 | \( 1 + 17.5iT - 529T^{2} \) |
| 29 | \( 1 - 8.83T + 841T^{2} \) |
| 31 | \( 1 + 0.583iT - 961T^{2} \) |
| 37 | \( 1 + 32.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 71.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.65iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 63.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 30.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 65.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 86.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 30T + 7.92e3T^{2} \) |
| 97 | \( 1 - 119.T + 9.40e3T^{2} \) |
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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
−8.633278859699991766334333267785, −7.77940626918198880246201862086, −6.75818082173818434800161497820, −5.98492732506333217125146088584, −5.41826957169619673233762253611, −4.74575524564694784305615760567, −3.27681263008390109319308627779, −2.75341390025117576139904833695, −1.72940667944930734573500459277, −0.31732520039900472555541712361,
1.13463929813264346313486635923, 1.98231756731461471763614562058, 3.24415497636741171820031022944, 4.07855961333512989684150550285, 4.89232313935445178910325003664, 5.60513370487786505120344875815, 6.77122004767835678121866621341, 7.29341373704611994419154985049, 7.75866602233643866132774378235, 8.829554142162266872386644221708
