L(s) = 1 | + (1 + i)2-s + (−2.42 + 2.42i)3-s + 2i·4-s + (1.75 + 1.39i)5-s − 4.84·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 8.74i·9-s + (0.359 + 3.14i)10-s + (−4.84 − 4.84i)12-s + (2.42 − 2.42i)13-s + 3.74i·14-s + (−7.61 + 0.870i)15-s − 4·16-s + (8.74 − 8.74i)18-s − 4.21i·19-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.39 + 1.39i)3-s + i·4-s + (0.782 + 0.622i)5-s − 1.97·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.91i·9-s + (0.113 + 0.993i)10-s + (−1.39 − 1.39i)12-s + (0.672 − 0.672i)13-s + 0.999i·14-s + (−1.96 + 0.224i)15-s − 16-s + (2.06 − 2.06i)18-s − 0.968i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0811829 + 1.37527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0811829 + 1.37527i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (-1.75 - 1.39i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 3 | \( 1 + (2.42 - 2.42i)T - 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-2.42 + 2.42i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 4.21iT - 19T^{2} \) |
| 23 | \( 1 + (-0.741 + 0.741i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 0.625iT - 59T^{2} \) |
| 61 | \( 1 - 5.47T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 15.7iT - 79T^{2} \) |
| 83 | \( 1 + (11.4 - 11.4i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−12.14509733948928736686135966928, −11.26961524418877315615202257169, −10.73489063222931068770885535955, −9.549970706343219264932534661050, −8.630026955860826606140864758896, −6.89900917186740086309776085089, −5.92951617809443127690396394538, −5.40707817928963501961833191065, −4.47068490601417433318671083236, −3.10049946236278568573381957593,
1.12240344117003200361874487833, 1.92617883642281988941808591821, 4.41053986502618376954193049358, 5.39638898602376712338122004248, 6.13530400647983898045817669117, 7.08357415664989856595334780411, 8.397096428340398748473920468525, 10.01554175262039064966226288245, 10.87589736272933343342835109926, 11.56633868409297619861498466584