| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s + 10-s − 2·11-s + 6·13-s − 16-s − 6·17-s − 6·19-s − 20-s − 2·22-s − 4·23-s + 25-s + 6·26-s + 8·29-s − 6·31-s + 5·32-s − 6·34-s − 6·37-s − 6·38-s − 3·40-s − 6·41-s + 2·44-s − 4·46-s + 50-s − 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s − 0.603·11-s + 1.66·13-s − 1/4·16-s − 1.45·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s + 1.48·29-s − 1.07·31-s + 0.883·32-s − 1.02·34-s − 0.986·37-s − 0.973·38-s − 0.474·40-s − 0.937·41-s + 0.301·44-s − 0.589·46-s + 0.141·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.664068779183690982679789701928, −8.187515450479836121328926189262, −6.72192713162943308739304986767, −6.22667438993578767114819025037, −5.47069050468901015919912284150, −4.53481759613044841649719781882, −3.92378848482334656823977786158, −2.89806695355720571072496721431, −1.76173301619082552236303695523, 0,
1.76173301619082552236303695523, 2.89806695355720571072496721431, 3.92378848482334656823977786158, 4.53481759613044841649719781882, 5.47069050468901015919912284150, 6.22667438993578767114819025037, 6.72192713162943308739304986767, 8.187515450479836121328926189262, 8.664068779183690982679789701928
