| L(s) = 1 | − 2·5-s − 4·7-s − 3·9-s − 2·13-s + 17-s + 4·19-s − 4·23-s − 25-s + 6·29-s − 4·31-s + 8·35-s − 2·37-s − 6·41-s − 4·43-s + 6·45-s + 9·49-s + 6·53-s + 12·59-s − 10·61-s + 12·63-s + 4·65-s − 4·67-s + 4·71-s − 6·73-s − 12·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 9-s − 0.554·13-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s − 0.937·41-s − 0.609·43-s + 0.894·45-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s + 1.51·63-s + 0.496·65-s − 0.488·67-s + 0.474·71-s − 0.702·73-s − 1.35·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−11.77779497781898747190652445510, −10.40227615321498563936322188024, −9.578693167598756855048719447448, −8.535338766321817121127070704904, −7.50901244752622102880251473780, −6.48235693417520450637241002127, −5.37611913672208711896121353581, −3.78911546643884845747068338400, −2.88331668043848910381979864823, 0,
2.88331668043848910381979864823, 3.78911546643884845747068338400, 5.37611913672208711896121353581, 6.48235693417520450637241002127, 7.50901244752622102880251473780, 8.535338766321817121127070704904, 9.578693167598756855048719447448, 10.40227615321498563936322188024, 11.77779497781898747190652445510
