L(s) = 1 | − 5-s − 2·7-s + 4·11-s − 13-s − 4·17-s − 6·19-s + 25-s − 4·29-s + 10·31-s + 2·35-s − 2·37-s − 6·41-s + 8·43-s + 8·47-s − 3·49-s − 4·53-s − 4·55-s − 12·59-s + 2·61-s + 65-s + 10·67-s − 6·73-s − 8·77-s − 12·79-s + 4·83-s + 4·85-s + 14·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.277·13-s − 0.970·17-s − 1.37·19-s + 1/5·25-s − 0.742·29-s + 1.79·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s − 0.549·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.124·65-s + 1.22·67-s − 0.702·73-s − 0.911·77-s − 1.35·79-s + 0.439·83-s + 0.433·85-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.233318119\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233318119\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.65901110254070414864583726885, −6.84954393490990062905870693519, −6.46643342798898299806872852292, −5.87861199091569254401738631044, −4.67745542826184328258705718866, −4.24847390325441346659809529586, −3.52147510525138149487220575530, −2.65702625651464032987178901584, −1.76146820349443026374150294985, −0.52191500735629528567268755837,
0.52191500735629528567268755837, 1.76146820349443026374150294985, 2.65702625651464032987178901584, 3.52147510525138149487220575530, 4.24847390325441346659809529586, 4.67745542826184328258705718866, 5.87861199091569254401738631044, 6.46643342798898299806872852292, 6.84954393490990062905870693519, 7.65901110254070414864583726885