| L(s) = 1 | + 22·5-s + 7·7-s − 26·11-s − 54·13-s + 74·17-s − 116·19-s + 58·23-s + 359·25-s + 208·29-s + 252·31-s + 154·35-s + 50·37-s + 126·41-s − 164·43-s + 444·47-s + 49·49-s + 12·53-s − 572·55-s − 124·59-s − 162·61-s − 1.18e3·65-s + 860·67-s + 238·71-s − 146·73-s − 182·77-s + 984·79-s − 656·83-s + ⋯ |
| L(s) = 1 | + 1.96·5-s + 0.377·7-s − 0.712·11-s − 1.15·13-s + 1.05·17-s − 1.40·19-s + 0.525·23-s + 2.87·25-s + 1.33·29-s + 1.46·31-s + 0.743·35-s + 0.222·37-s + 0.479·41-s − 0.581·43-s + 1.37·47-s + 1/7·49-s + 0.0311·53-s − 1.40·55-s − 0.273·59-s − 0.340·61-s − 2.26·65-s + 1.56·67-s + 0.397·71-s − 0.234·73-s − 0.269·77-s + 1.40·79-s − 0.867·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.197829900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.197829900\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 5 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 208 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 50 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 444 T + p^{3} T^{2} \) |
| 53 | \( 1 - 12 T + p^{3} T^{2} \) |
| 59 | \( 1 + 124 T + p^{3} T^{2} \) |
| 61 | \( 1 + 162 T + p^{3} T^{2} \) |
| 67 | \( 1 - 860 T + p^{3} T^{2} \) |
| 71 | \( 1 - 238 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 984 T + p^{3} T^{2} \) |
| 83 | \( 1 + 656 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 - 526 T + p^{3} T^{2} \) |
| 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
−9.814120072565119013852371999090, −8.850048902239946680037130407444, −8.011286588855377335806403218319, −6.87671037241305453452974694257, −6.11290531430066851916910157565, −5.26230219301382659975179543305, −4.64000136345581432583032518571, −2.78108550828981610570345345014, −2.22670586856476110281884264682, −0.986458660726241516064056575761,
0.986458660726241516064056575761, 2.22670586856476110281884264682, 2.78108550828981610570345345014, 4.64000136345581432583032518571, 5.26230219301382659975179543305, 6.11290531430066851916910157565, 6.87671037241305453452974694257, 8.011286588855377335806403218319, 8.850048902239946680037130407444, 9.814120072565119013852371999090
