| L(s) = 1 | + 5-s − 2·7-s − 6·11-s + 4·13-s + 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s − 2·35-s − 8·37-s + 8·43-s − 3·49-s − 6·53-s − 6·55-s − 6·59-s − 2·61-s + 4·65-s − 4·67-s − 12·71-s − 10·73-s + 12·77-s + 4·79-s − 12·83-s + 6·85-s − 12·89-s − 8·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.80·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 1.31·37-s + 1.21·43-s − 3/7·49-s − 0.824·53-s − 0.809·55-s − 0.781·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s − 1.17·73-s + 1.36·77-s + 0.450·79-s − 1.31·83-s + 0.650·85-s − 1.27·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.330531568433955397282390142839, −7.73490121522356695217356839933, −6.86802385357238762255411937862, −5.84293355103774809066040280829, −5.62305662493911647485930324213, −4.48715934481021693344557314737, −3.37793372322877666574139691343, −2.75202257331423339947272598173, −1.53717311332283821140901478063, 0,
1.53717311332283821140901478063, 2.75202257331423339947272598173, 3.37793372322877666574139691343, 4.48715934481021693344557314737, 5.62305662493911647485930324213, 5.84293355103774809066040280829, 6.86802385357238762255411937862, 7.73490121522356695217356839933, 8.330531568433955397282390142839
