| L(s) = 1 | + 2.64i·7-s − 3·9-s + 5.29i·11-s + 5.29i·23-s + 5·25-s + 2·29-s − 6·37-s + 5.29i·43-s − 7.00·49-s + 10·53-s − 7.93i·63-s − 15.8i·67-s + 5.29i·71-s − 14.0·77-s − 15.8i·79-s + ⋯ |
| L(s) = 1 | + 0.999i·7-s − 9-s + 1.59i·11-s + 1.10i·23-s + 25-s + 0.371·29-s − 0.986·37-s + 0.806i·43-s − 49-s + 1.37·53-s − 0.999i·63-s − 1.93i·67-s + 0.627i·71-s − 1.59·77-s − 1.78i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.786169 + 0.786169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786169 + 0.786169i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.29iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.49152497528393041707668265840, −10.38635889052106819476988993967, −9.410178078317078973113001315154, −8.738955865111390081529635852526, −7.69854706366703164474507715074, −6.65260575479626587316422373006, −5.56225003010544887796105488184, −4.73502645778609063858650961966, −3.16007984510917038666590124866, −2.00144836748714336353360416757,
0.69463734471982938687345208467, 2.81427536528123045363032090073, 3.83073094205928889099629351890, 5.17334268337988932652495499792, 6.18572740184707529442717171464, 7.11618333570179855500660868961, 8.401683501285587307503468036059, 8.758309550910820857072502984613, 10.21859226079339160727226181413, 10.86788662483466188817600966342
