| L(s) = 1 | − i·3-s − 0.559·5-s − 0.446i·7-s − 9-s + 0.0695i·11-s − 0.0898i·13-s + 0.559i·15-s − 0.411i·17-s + 3.12i·19-s − 0.446·21-s − 4.96·23-s − 4.68·25-s + i·27-s + 0.146i·29-s + 4.06·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.250·5-s − 0.168i·7-s − 0.333·9-s + 0.0209i·11-s − 0.0249i·13-s + 0.144i·15-s − 0.0999i·17-s + 0.715i·19-s − 0.0973·21-s − 1.03·23-s − 0.937·25-s + 0.192i·27-s + 0.0272i·29-s + 0.730·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178030604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178030604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 41 | \( 1 + (-4.92 - 4.09i)T \) |
| good | 5 | \( 1 + 0.559T + 5T^{2} \) |
| 7 | \( 1 + 0.446iT - 7T^{2} \) |
| 11 | \( 1 - 0.0695iT - 11T^{2} \) |
| 13 | \( 1 + 0.0898iT - 13T^{2} \) |
| 17 | \( 1 + 0.411iT - 17T^{2} \) |
| 19 | \( 1 - 3.12iT - 19T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 - 0.146iT - 29T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 8.53iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.33T + 59T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 + 14.9iT - 67T^{2} \) |
| 71 | \( 1 - 2.78iT - 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 9.92T + 83T^{2} \) |
| 89 | \( 1 + 6.90iT - 89T^{2} \) |
| 97 | \( 1 - 13.6iT - 97T^{2} \) |
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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
−8.275620720391447615561796288306, −7.86853436047908231397723809998, −7.25737589909590917957824314681, −6.16746950116799886132605468945, −5.94277222661695153901126887442, −4.70292468344313494910686438625, −4.00723781973914445498456358344, −3.02532685093692588904449674429, −2.07213451190788589678410927775, −1.01106256319104898288422094784,
0.38699628953819191753638964471, 1.95978349008157943449636978687, 2.89220139910887993748590676025, 3.87749811053579215393354185789, 4.44246034820812221143605692318, 5.37982101421445904390883144386, 6.04472443034500234006514387414, 6.88436203265020212444531135178, 7.75334165514661992734946290161, 8.404286152951127243566916335316
