| L(s) = 1 | − 5-s + 4·7-s − 2·13-s − 2·17-s + 4·19-s + 25-s − 6·29-s − 8·31-s − 4·35-s + 6·37-s − 10·41-s − 43-s + 9·49-s − 6·53-s − 2·61-s + 2·65-s + 12·67-s − 8·71-s − 2·73-s + 8·79-s + 4·83-s + 2·85-s + 6·89-s − 8·91-s − 4·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s − 1.56·41-s − 0.152·43-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.635·89-s − 0.838·91-s − 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.84285547052407, −13.24445233960101, −12.87187716566815, −12.20517123194327, −11.72272951912537, −11.39498458461826, −11.03259548808990, −10.49387327728018, −9.918051820086505, −9.165945202970736, −9.050258990740353, −8.179909886093930, −7.881583531446211, −7.477750230073231, −6.965313117843807, −6.348892693617879, −5.452244387372004, −5.260724592223630, −4.685266490592944, −4.145755837495468, −3.541683742649451, −2.926771005350920, −1.998826924080932, −1.761436985073483, −0.8748066432862999, 0,
0.8748066432862999, 1.761436985073483, 1.998826924080932, 2.926771005350920, 3.541683742649451, 4.145755837495468, 4.685266490592944, 5.260724592223630, 5.452244387372004, 6.348892693617879, 6.965313117843807, 7.477750230073231, 7.881583531446211, 8.179909886093930, 9.050258990740353, 9.165945202970736, 9.918051820086505, 10.49387327728018, 11.03259548808990, 11.39498458461826, 11.72272951912537, 12.20517123194327, 12.87187716566815, 13.24445233960101, 13.84285547052407
