| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 2·5-s + 4·6-s + 3·7-s + 9-s − 4·10-s + 11-s − 4·12-s − 2·13-s − 6·14-s − 4·15-s − 4·16-s + 17-s − 2·18-s − 8·19-s + 4·20-s − 6·21-s − 2·22-s − 4·23-s − 25-s + 4·26-s + 4·27-s + 6·28-s − 3·29-s + 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s + 1.13·7-s + 1/3·9-s − 1.26·10-s + 0.301·11-s − 1.15·12-s − 0.554·13-s − 1.60·14-s − 1.03·15-s − 16-s + 0.242·17-s − 0.471·18-s − 1.83·19-s + 0.894·20-s − 1.30·21-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 0.784·26-s + 0.769·27-s + 1.13·28-s − 0.557·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.75494837139515219968341290557, −6.75012552165660343973192625820, −6.19055357550393324640709197525, −5.65598534290878769128784277857, −4.66363809259899346878807581481, −4.32006235456487148017071771912, −2.50367436275456674849636442653, −1.88556142739126370734566709102, −1.05767406365510219838793062637, 0,
1.05767406365510219838793062637, 1.88556142739126370734566709102, 2.50367436275456674849636442653, 4.32006235456487148017071771912, 4.66363809259899346878807581481, 5.65598534290878769128784277857, 6.19055357550393324640709197525, 6.75012552165660343973192625820, 7.75494837139515219968341290557
