| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 11-s + 13-s − 15-s + 17-s − 7·19-s − 3·21-s + 6·23-s + 25-s − 27-s − 5·29-s + 2·31-s − 33-s + 3·35-s + 11·37-s − 39-s − 11·41-s − 2·43-s + 45-s − 5·47-s + 2·49-s − 51-s + 53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.60·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.174·33-s + 0.507·35-s + 1.80·37-s − 0.160·39-s − 1.71·41-s − 0.304·43-s + 0.149·45-s − 0.729·47-s + 2/7·49-s − 0.140·51-s + 0.137·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.26640172181891, −14.98857068934134, −14.72716369427019, −13.92920853647324, −13.32054180042525, −12.97499537047314, −12.31627987150224, −11.66414341409686, −11.23088212826881, −10.78628829850246, −10.30152859558162, −9.552107528576438, −9.009194148974741, −8.363156728532857, −7.915336695536890, −7.127023270122787, −6.554088385235519, −6.004749977711618, −5.387398539064729, −4.679531895010055, −4.378700834576329, −3.429943133418470, −2.551207722759845, −1.677527366036413, −1.243337511262271, 0,
1.243337511262271, 1.677527366036413, 2.551207722759845, 3.429943133418470, 4.378700834576329, 4.679531895010055, 5.387398539064729, 6.004749977711618, 6.554088385235519, 7.127023270122787, 7.915336695536890, 8.363156728532857, 9.009194148974741, 9.552107528576438, 10.30152859558162, 10.78628829850246, 11.23088212826881, 11.66414341409686, 12.31627987150224, 12.97499537047314, 13.32054180042525, 13.92920853647324, 14.72716369427019, 14.98857068934134, 15.26640172181891
