| L(s) = 1 | − i·3-s + (2.22 − 0.254i)5-s + 2.64i·7-s − 9-s − 1.51·11-s − 3.87i·13-s + (−0.254 − 2.22i)15-s − 3.31i·17-s + 7.08·19-s + 2.64·21-s − 4.82i·23-s + (4.87 − 1.12i)25-s + i·27-s − 2.18·29-s − 7.36·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.993 − 0.113i)5-s + 0.998i·7-s − 0.333·9-s − 0.456·11-s − 1.07i·13-s + (−0.0656 − 0.573i)15-s − 0.803i·17-s + 1.62·19-s + 0.576·21-s − 1.00i·23-s + (0.974 − 0.225i)25-s + 0.192i·27-s − 0.405·29-s − 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047376949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047376949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.22 + 0.254i)T \) |
| good | 7 | \( 1 - 2.64iT - 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 3.87iT - 13T^{2} \) |
| 17 | \( 1 + 3.31iT - 17T^{2} \) |
| 19 | \( 1 - 7.08T + 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.87iT - 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 1.01iT - 43T^{2} \) |
| 47 | \( 1 - 7.08iT - 47T^{2} \) |
| 53 | \( 1 - 4.50iT - 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 1.01iT - 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 + 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.309471573684089396183034577504, −7.56670676359495929136954560756, −6.88262497655913215692450103816, −5.84754103947972421134877980093, −5.50910260064004046251643484600, −4.91400424913137845578546006662, −3.29947706823111444104033764028, −2.64598945939180425451645859963, −1.85960592357421861478076960784, −0.60985970578617965420753781680,
1.23493145216200480700154482240, 2.14591509078287772576785165050, 3.42378526854679473334666466122, 3.89514485867190699473626192229, 5.16858138796909179495115098351, 5.37276395453345559923235943541, 6.54704702523888695218030105375, 7.07219689754410465817041415024, 7.906984142259770485180576296831, 8.824060620896358319981304739328
