| L(s) = 1 | − 2·5-s − 7-s + 11-s − 6·13-s − 2·17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 2·35-s − 6·37-s − 10·41-s − 4·43-s − 8·47-s + 49-s + 6·53-s − 2·55-s − 4·59-s + 10·61-s + 12·65-s − 12·67-s + 2·73-s − 77-s − 16·79-s − 4·83-s + 4·85-s − 18·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 1.28·61-s + 1.48·65-s − 1.46·67-s + 0.234·73-s − 0.113·77-s − 1.80·79-s − 0.439·83-s + 0.433·85-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.31255740619993, −14.61429140677682, −14.34206492428631, −13.55734961876357, −13.11853237379457, −12.45944088163424, −12.09151653041275, −11.53123119408761, −11.29406261683786, −10.35655581350758, −9.946264036372715, −9.497441719093682, −8.805668579897763, −8.350693265528723, −7.583263270589660, −7.131623057885274, −6.934018053037100, −5.977959703130231, −5.278000924410227, −4.869995592498266, −4.115215677344699, −3.525842316449838, −3.012840999200863, −2.153277269614215, −1.436935358021841, 0, 0,
1.436935358021841, 2.153277269614215, 3.012840999200863, 3.525842316449838, 4.115215677344699, 4.869995592498266, 5.278000924410227, 5.977959703130231, 6.934018053037100, 7.131623057885274, 7.583263270589660, 8.350693265528723, 8.805668579897763, 9.497441719093682, 9.946264036372715, 10.35655581350758, 11.29406261683786, 11.53123119408761, 12.09151653041275, 12.45944088163424, 13.11853237379457, 13.55734961876357, 14.34206492428631, 14.61429140677682, 15.31255740619993
