| L(s) = 1 | + i·3-s − 2.79i·5-s − 1.28·7-s − 9-s + 1.50i·11-s − i·13-s + 2.79·15-s − 5.80·17-s − 0.0480i·19-s − 1.28i·21-s + 1.11·23-s − 2.80·25-s − i·27-s − 9.39i·29-s − 6.32·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.24i·5-s − 0.485·7-s − 0.333·9-s + 0.455i·11-s − 0.277i·13-s + 0.721·15-s − 1.40·17-s − 0.0110i·19-s − 0.280i·21-s + 0.232·23-s − 0.560·25-s − 0.192i·27-s − 1.74i·29-s − 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4030838919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4030838919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.79iT - 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 1.50iT - 11T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 0.0480iT - 19T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 + 9.39iT - 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 2.70iT - 37T^{2} \) |
| 41 | \( 1 + 6.18T + 41T^{2} \) |
| 43 | \( 1 - 7.57iT - 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 3.90iT - 53T^{2} \) |
| 59 | \( 1 + 6.54iT - 59T^{2} \) |
| 61 | \( 1 + 2.76iT - 61T^{2} \) |
| 67 | \( 1 + 4.04iT - 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 - 0.863T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 5.11iT - 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−9.463086776265626211578698323228, −8.636429738117454028569293109530, −7.980373020110242745176478692220, −6.77912244556199875293206999247, −5.90505905595725447533099623373, −4.83912730914904686629620809199, −4.41179494363706817554809483098, −3.21845711172931379578167262304, −1.84634649840285564243833867598, −0.15849126424939616790988588424,
1.84232965219323793639876208142, 2.92770094320069295155558313591, 3.67561380957610362619106496124, 5.08628650181498001641765696338, 6.17973891985686339164861090568, 6.86823651768640321695626456199, 7.22753989460303540096628791960, 8.490702297008216872941108832645, 9.110325867935258232036396009473, 10.18589042213334775198226746733
