| L(s) = 1 | + 5.61·5-s − 8.82·7-s − 13.4·11-s − 2.26·17-s − 19·19-s + 34.8·23-s + 6.52·25-s − 49.5·35-s − 31.1·43-s + 93.2·47-s + 28.8·49-s − 75.7·55-s + 108.·61-s + 137.·73-s + 119.·77-s + 139.·83-s − 12.7·85-s − 106.·95-s + 174.·101-s + 195.·115-s + 19.9·119-s + ⋯ |
| L(s) = 1 | + 1.12·5-s − 1.26·7-s − 1.22·11-s − 0.133·17-s − 19-s + 1.51·23-s + 0.261·25-s − 1.41·35-s − 0.725·43-s + 1.98·47-s + 0.589·49-s − 1.37·55-s + 1.77·61-s + 1.87·73-s + 1.54·77-s + 1.68·83-s − 0.149·85-s − 1.12·95-s + 1.72·101-s + 1.70·115-s + 0.168·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.726383164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726383164\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 5.61T + 25T^{2} \) |
| 7 | \( 1 + 8.82T + 49T^{2} \) |
| 11 | \( 1 + 13.4T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 2.26T + 289T^{2} \) |
| 23 | \( 1 - 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 93.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
| 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
−8.876732764348939430530935308815, −7.894234215387570755754266934566, −6.91649389184519195729429387632, −6.38909822997245689082318828669, −5.58818985207763691702844009466, −4.95572330341781161657182681700, −3.72389030326238303386969662419, −2.75453950399730291729078369312, −2.13902564951818936750599917585, −0.62734886380083067181166422519,
0.62734886380083067181166422519, 2.13902564951818936750599917585, 2.75453950399730291729078369312, 3.72389030326238303386969662419, 4.95572330341781161657182681700, 5.58818985207763691702844009466, 6.38909822997245689082318828669, 6.91649389184519195729429387632, 7.894234215387570755754266934566, 8.876732764348939430530935308815
