| L(s) = 1 | − 4·7-s − 13-s + 3·19-s − 4·23-s − 29-s − 8·31-s − 37-s − 41-s + 6·43-s − 11·47-s + 9·49-s + 3·53-s + 10·59-s − 4·61-s + 13·67-s − 9·71-s − 3·79-s − 2·83-s − 10·89-s + 4·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s + 0.688·19-s − 0.834·23-s − 0.185·29-s − 1.43·31-s − 0.164·37-s − 0.156·41-s + 0.914·43-s − 1.60·47-s + 9/7·49-s + 0.412·53-s + 1.30·59-s − 0.512·61-s + 1.58·67-s − 1.06·71-s − 0.337·79-s − 0.219·83-s − 1.05·89-s + 0.419·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.25341591324386, −12.81217200735330, −12.61588983715347, −11.95513893395711, −11.55514274619044, −11.02805136439749, −10.40238860041749, −9.985828903861143, −9.589614683214192, −9.256538675130148, −8.653769881865116, −8.119780681603363, −7.483900590778185, −7.066409879129172, −6.663725410509158, −6.020568439311407, −5.655530352459832, −5.137341919174235, −4.391533961874549, −3.769998024530735, −3.420617162488824, −2.844362160775049, −2.221260061354541, −1.558239366231542, −0.6306208069484096, 0,
0.6306208069484096, 1.558239366231542, 2.221260061354541, 2.844362160775049, 3.420617162488824, 3.769998024530735, 4.391533961874549, 5.137341919174235, 5.655530352459832, 6.020568439311407, 6.663725410509158, 7.066409879129172, 7.483900590778185, 8.119780681603363, 8.653769881865116, 9.256538675130148, 9.589614683214192, 9.985828903861143, 10.40238860041749, 11.02805136439749, 11.55514274619044, 11.95513893395711, 12.61588983715347, 12.81217200735330, 13.25341591324386
