| L(s) = 1 | + 136.·3-s + 2.55e3·5-s − 9.39e3·7-s − 1.04e3·9-s − 4.40e4·11-s + 2.85e4·13-s + 3.48e5·15-s + 2.82e4·17-s − 2.73e5·19-s − 1.28e6·21-s + 1.12e6·23-s + 4.57e6·25-s − 2.82e6·27-s − 1.63e6·29-s − 6.65e6·31-s − 6.02e6·33-s − 2.40e7·35-s − 1.71e7·37-s + 3.89e6·39-s − 5.15e6·41-s + 1.97e7·43-s − 2.66e6·45-s − 4.82e7·47-s + 4.80e7·49-s + 3.86e6·51-s − 3.06e7·53-s − 1.12e8·55-s + ⋯ |
| L(s) = 1 | + 0.973·3-s + 1.82·5-s − 1.47·7-s − 0.0529·9-s − 0.908·11-s + 0.277·13-s + 1.77·15-s + 0.0821·17-s − 0.482·19-s − 1.44·21-s + 0.841·23-s + 2.34·25-s − 1.02·27-s − 0.429·29-s − 1.29·31-s − 0.883·33-s − 2.70·35-s − 1.50·37-s + 0.269·39-s − 0.284·41-s + 0.879·43-s − 0.0967·45-s − 1.44·47-s + 1.18·49-s + 0.0799·51-s − 0.533·53-s − 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 136.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.40e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.82e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.63e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.15e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.48e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9T + 7.60e17T^{2} \) |
| 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
−9.975014240838993116974709331626, −9.316472949769523351977604916039, −8.621735814681461120399683583853, −7.11327692946556429761837346023, −6.11085580338786371832332775452, −5.30524403864015557455964673795, −3.38678278814354224273486520182, −2.66579865352240529894891561831, −1.73102366334286269743387912110, 0,
1.73102366334286269743387912110, 2.66579865352240529894891561831, 3.38678278814354224273486520182, 5.30524403864015557455964673795, 6.11085580338786371832332775452, 7.11327692946556429761837346023, 8.621735814681461120399683583853, 9.316472949769523351977604916039, 9.975014240838993116974709331626
