| L(s) = 1 | + 5-s + 7-s − 3·9-s + 6·11-s − 2·17-s − 4·19-s − 23-s + 25-s − 2·29-s + 2·31-s + 35-s + 8·37-s + 6·41-s − 3·45-s + 4·47-s + 49-s − 12·53-s + 6·55-s − 10·59-s + 2·61-s − 3·63-s + 4·67-s − 12·71-s − 8·73-s + 6·77-s + 10·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s + 1.80·11-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.169·35-s + 1.31·37-s + 0.937·41-s − 0.447·45-s + 0.583·47-s + 1/7·49-s − 1.64·53-s + 0.809·55-s − 1.30·59-s + 0.256·61-s − 0.377·63-s + 0.488·67-s − 1.42·71-s − 0.936·73-s + 0.683·77-s + 1.12·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.69326442791498, −14.20961143673495, −13.91895293988082, −13.28143001691189, −12.60733443518742, −12.23436112213607, −11.40929045937447, −11.36267812662822, −10.75756218465398, −10.05824928797018, −9.334672678762489, −9.111769096098104, −8.589799796910067, −7.991738287258224, −7.380445333217077, −6.570439448560371, −6.207741553967214, −5.862169557472082, −5.010422256875084, −4.310861430703794, −3.989729700657816, −3.061014335841796, −2.465450101970274, −1.716452031279683, −1.078722848944648, 0,
1.078722848944648, 1.716452031279683, 2.465450101970274, 3.061014335841796, 3.989729700657816, 4.310861430703794, 5.010422256875084, 5.862169557472082, 6.207741553967214, 6.570439448560371, 7.380445333217077, 7.991738287258224, 8.589799796910067, 9.111769096098104, 9.334672678762489, 10.05824928797018, 10.75756218465398, 11.36267812662822, 11.40929045937447, 12.23436112213607, 12.60733443518742, 13.28143001691189, 13.91895293988082, 14.20961143673495, 14.69326442791498
