| L(s) = 1 | − 2·5-s + 4·11-s + 6·13-s − 4·17-s + 6·19-s + 4·23-s − 25-s − 6·29-s + 4·31-s + 6·37-s + 4·41-s − 12·43-s − 12·47-s + 6·53-s − 8·55-s − 6·59-s − 6·61-s − 12·65-s − 12·67-s − 8·71-s − 6·83-s + 8·85-s − 16·89-s − 12·95-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 1.66·13-s − 0.970·17-s + 1.37·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.624·41-s − 1.82·43-s − 1.75·47-s + 0.824·53-s − 1.07·55-s − 0.781·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s − 0.949·71-s − 0.658·83-s + 0.867·85-s − 1.69·89-s − 1.23·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.34755846251322, −15.13562915279668, −14.49793006114185, −13.75604496012206, −13.38750495144183, −12.96698730368697, −12.06981113145555, −11.58866668229686, −11.32875758536763, −10.91793038241248, −9.992245390283334, −9.440525749971564, −8.831459408744054, −8.497221375828278, −7.719742100245167, −7.273730847793301, −6.470663637376625, −6.186788358241313, −5.335670685435215, −4.526560196132558, −4.014568836505442, −3.445869770715749, −2.901334472850476, −1.584322565553055, −1.168252463241035, 0,
1.168252463241035, 1.584322565553055, 2.901334472850476, 3.445869770715749, 4.014568836505442, 4.526560196132558, 5.335670685435215, 6.186788358241313, 6.470663637376625, 7.273730847793301, 7.719742100245167, 8.497221375828278, 8.831459408744054, 9.440525749971564, 9.992245390283334, 10.91793038241248, 11.32875758536763, 11.58866668229686, 12.06981113145555, 12.96698730368697, 13.38750495144183, 13.75604496012206, 14.49793006114185, 15.13562915279668, 15.34755846251322
