| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 4·13-s + 15-s + 6·17-s − 7·19-s − 21-s + 23-s + 25-s + 27-s − 6·29-s − 4·31-s + 33-s − 35-s − 2·37-s + 4·39-s + 9·41-s − 2·43-s + 45-s − 7·47-s + 49-s + 6·51-s + 5·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s − 1.60·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.640·39-s + 1.40·41-s − 0.304·43-s + 0.149·45-s − 1.02·47-s + 1/7·49-s + 0.840·51-s + 0.686·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.93511576479204, −14.56039214438602, −14.14528474572772, −13.46034490674110, −13.00196476447760, −12.72210071557633, −12.03951375447487, −11.38722209655384, −10.63501804188764, −10.50556075953622, −9.654150265823990, −9.214698225317964, −8.821576115729321, −8.100215173497250, −7.709536171361504, −6.895569850629071, −6.426968494536313, −5.770714073779048, −5.386647485341800, −4.307814741239914, −3.907805732926482, −3.241779449563872, −2.634572503457242, −1.708818998619257, −1.265768843119586, 0,
1.265768843119586, 1.708818998619257, 2.634572503457242, 3.241779449563872, 3.907805732926482, 4.307814741239914, 5.386647485341800, 5.770714073779048, 6.426968494536313, 6.895569850629071, 7.709536171361504, 8.100215173497250, 8.821576115729321, 9.214698225317964, 9.654150265823990, 10.50556075953622, 10.63501804188764, 11.38722209655384, 12.03951375447487, 12.72210071557633, 13.00196476447760, 13.46034490674110, 14.14528474572772, 14.56039214438602, 14.93511576479204
