| L(s) = 1 | − 7-s − 11-s − 4·13-s − 5·17-s + 19-s − 23-s + 29-s − 2·31-s − 4·37-s + 6·41-s − 5·43-s + 8·47-s + 49-s + 3·53-s + 7·59-s + 3·61-s + 4·67-s + 4·73-s + 77-s − 10·79-s + 3·83-s + 13·89-s + 4·91-s − 9·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s − 1.10·13-s − 1.21·17-s + 0.229·19-s − 0.208·23-s + 0.185·29-s − 0.359·31-s − 0.657·37-s + 0.937·41-s − 0.762·43-s + 1.16·47-s + 1/7·49-s + 0.412·53-s + 0.911·59-s + 0.384·61-s + 0.488·67-s + 0.468·73-s + 0.113·77-s − 1.12·79-s + 0.329·83-s + 1.37·89-s + 0.419·91-s − 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.35891800113724, −13.93136303488392, −13.34061788399608, −12.89006702461180, −12.49268895408639, −11.83520058027664, −11.52484833435238, −10.74127391789221, −10.43703886379369, −9.797517620620316, −9.381087301123176, −8.802249264920772, −8.339180587328856, −7.582535709391210, −7.202212916517713, −6.669460814358616, −6.112864774291749, −5.358758565423203, −5.009636260529386, −4.245970296697837, −3.796373539062323, −2.925646678774508, −2.414677401181319, −1.864197537153860, −0.7610500790729197, 0,
0.7610500790729197, 1.864197537153860, 2.414677401181319, 2.925646678774508, 3.796373539062323, 4.245970296697837, 5.009636260529386, 5.358758565423203, 6.112864774291749, 6.669460814358616, 7.202212916517713, 7.582535709391210, 8.339180587328856, 8.802249264920772, 9.381087301123176, 9.797517620620316, 10.43703886379369, 10.74127391789221, 11.52484833435238, 11.83520058027664, 12.49268895408639, 12.89006702461180, 13.34061788399608, 13.93136303488392, 14.35891800113724
