| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 2·13-s + 15-s − 6·17-s − 4·19-s + 4·21-s − 23-s + 25-s − 27-s − 29-s + 4·31-s + 4·35-s + 6·37-s + 2·39-s − 2·41-s − 4·43-s − 45-s − 8·47-s + 9·49-s + 6·51-s + 10·53-s + 4·57-s − 10·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s + 0.718·31-s + 0.676·35-s + 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.47090142398688, −13.05189307131485, −12.48955505236079, −12.18492760624621, −11.63134091998999, −11.20056214078441, −10.55292774280415, −10.31511378794666, −9.670383927600652, −9.307072750953752, −8.767602219532655, −8.226200044803962, −7.624158448741702, −6.989964385088748, −6.642206561918448, −6.253603478939022, −5.820402114316169, −4.957462716969220, −4.563071473297381, −4.017037273003586, −3.491788968564770, −2.744995136165288, −2.354607235623152, −1.486088977120375, −0.4696300876350462, 0,
0.4696300876350462, 1.486088977120375, 2.354607235623152, 2.744995136165288, 3.491788968564770, 4.017037273003586, 4.563071473297381, 4.957462716969220, 5.820402114316169, 6.253603478939022, 6.642206561918448, 6.989964385088748, 7.624158448741702, 8.226200044803962, 8.767602219532655, 9.307072750953752, 9.670383927600652, 10.31511378794666, 10.55292774280415, 11.20056214078441, 11.63134091998999, 12.18492760624621, 12.48955505236079, 13.05189307131485, 13.47090142398688
