L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 4·11-s − 4·13-s + 15-s + 4·17-s − 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 4·33-s − 4·35-s − 2·37-s − 4·39-s + 10·41-s + 43-s + 45-s + 4·47-s + 9·49-s + 4·51-s + 2·53-s + 4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 0.258·15-s + 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.152·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.509547582\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.509547582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.79861787702835, −14.39795233041108, −13.66676048970316, −13.09446109284894, −12.75144100981251, −12.38689152431648, −11.72770510646580, −11.01765326927353, −10.38467378496242, −9.840065033539233, −9.485392492069058, −8.949274575540536, −8.720036729584638, −7.548604975370793, −7.181602891463600, −6.819575219576973, −5.963323446810806, −5.702608249166092, −4.747292849274687, −4.070143628754230, −3.478043596474111, −2.915917140212847, −2.316728426103992, −1.456488730307267, −0.5590689925858802,
0.5590689925858802, 1.456488730307267, 2.316728426103992, 2.915917140212847, 3.478043596474111, 4.070143628754230, 4.747292849274687, 5.702608249166092, 5.963323446810806, 6.819575219576973, 7.181602891463600, 7.548604975370793, 8.720036729584638, 8.949274575540536, 9.485392492069058, 9.840065033539233, 10.38467378496242, 11.01765326927353, 11.72770510646580, 12.38689152431648, 12.75144100981251, 13.09446109284894, 13.66676048970316, 14.39795233041108, 14.79861787702835