L(s) = 1 | + 5-s + 7-s + 3·13-s − 7·17-s + 4·19-s − 8·23-s − 4·25-s + 29-s − 8·31-s + 35-s − 9·37-s − 10·41-s − 4·43-s − 8·47-s + 49-s + 10·53-s + 12·59-s − 5·61-s + 3·65-s + 12·67-s + 7·73-s − 8·79-s − 7·85-s − 7·89-s + 3·91-s + 4·95-s − 14·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.832·13-s − 1.69·17-s + 0.917·19-s − 1.66·23-s − 4/5·25-s + 0.185·29-s − 1.43·31-s + 0.169·35-s − 1.47·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.640·61-s + 0.372·65-s + 1.46·67-s + 0.819·73-s − 0.900·79-s − 0.759·85-s − 0.741·89-s + 0.314·91-s + 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−8.202566200628385014565584731695, −7.09902371351759420493908756124, −6.59382405155590430851194316392, −5.68300664232913614006199108702, −5.15798081644993133919008038228, −4.09047037516015457565797117830, −3.50127719582088934517193564860, −2.17507434808660816022886144688, −1.62948929630295331041034629430, 0,
1.62948929630295331041034629430, 2.17507434808660816022886144688, 3.50127719582088934517193564860, 4.09047037516015457565797117830, 5.15798081644993133919008038228, 5.68300664232913614006199108702, 6.59382405155590430851194316392, 7.09902371351759420493908756124, 8.202566200628385014565584731695