| L(s) = 1 | + 5-s + 2·7-s + 2·13-s + 2·17-s − 2·23-s + 25-s + 4·29-s + 4·31-s + 2·35-s + 2·37-s − 4·41-s − 10·43-s + 6·47-s − 3·49-s − 6·53-s + 4·59-s − 10·61-s + 2·65-s − 4·67-s − 4·71-s − 2·73-s − 4·79-s − 10·83-s + 2·85-s + 4·89-s + 4·91-s + 18·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.554·13-s + 0.485·17-s − 0.417·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.624·41-s − 1.52·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s − 0.474·71-s − 0.234·73-s − 0.450·79-s − 1.09·83-s + 0.216·85-s + 0.423·89-s + 0.419·91-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.193897937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.193897937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.98139859064687, −12.25163369022049, −11.77947206810365, −11.60653431983671, −10.93964489106683, −10.40579441729549, −10.17630063088400, −9.631261728062741, −9.022539783865541, −8.629226069058705, −8.104677975957832, −7.806024719320556, −7.175873882630842, −6.562005822742956, −6.186685202236693, −5.693290040197889, −5.084558207418283, −4.690744486348313, −4.185770089405156, −3.465257438908138, −2.990732414062984, −2.374098394899627, −1.585733359008864, −1.366272711146465, −0.4830757545016100,
0.4830757545016100, 1.366272711146465, 1.585733359008864, 2.374098394899627, 2.990732414062984, 3.465257438908138, 4.185770089405156, 4.690744486348313, 5.084558207418283, 5.693290040197889, 6.186685202236693, 6.562005822742956, 7.175873882630842, 7.806024719320556, 8.104677975957832, 8.629226069058705, 9.022539783865541, 9.631261728062741, 10.17630063088400, 10.40579441729549, 10.93964489106683, 11.60653431983671, 11.77947206810365, 12.25163369022049, 12.98139859064687
