| L(s) = 1 | + (−0.292 − 1.70i)3-s + 3.41i·5-s − i·7-s + (−2.82 + i)9-s − 2·11-s + 3.41·13-s + (5.82 − i)15-s + 4.82i·17-s + 6.24i·19-s + (−1.70 + 0.292i)21-s + 4·23-s − 6.65·25-s + (2.53 + 4.53i)27-s + 4.82i·29-s − 1.17i·31-s + ⋯ |
| L(s) = 1 | + (−0.169 − 0.985i)3-s + 1.52i·5-s − 0.377i·7-s + (−0.942 + 0.333i)9-s − 0.603·11-s + 0.946·13-s + (1.50 − 0.258i)15-s + 1.17i·17-s + 1.43i·19-s + (−0.372 + 0.0639i)21-s + 0.834·23-s − 1.33·25-s + (0.487 + 0.872i)27-s + 0.896i·29-s − 0.210i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05658 + 0.546927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05658 + 0.546927i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.292 + 1.70i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 - 6.24iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.82iT - 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 9.31iT - 67T^{2} \) |
| 71 | \( 1 - 0.343T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−10.80555266010315370079183318194, −10.12279875913702050195185191247, −8.629030021855511485421250317282, −7.83217065478206207569347636582, −7.08118385336549422324909323453, −6.30727769041174894730966100778, −5.60962065453218825339127858738, −3.79791086170925282233638470151, −2.88018013703640576032443937339, −1.57797449405011527201244697217,
0.67680233468151085231554848849, 2.68225124675847116219726995264, 4.02368219379534441020657121560, 5.10664896202658723678167964576, 5.29865505945942911404291085520, 6.69862903310707563641387628365, 8.167046090181951819675347406988, 8.823094579369958037426166274041, 9.353578818386803860131134475006, 10.23922054085028677562014067114
