| L(s) = 1 | + 7-s + 4·11-s − 3·13-s − 7·17-s + 6·19-s − 9·23-s + 3·29-s − 7·31-s − 10·37-s + 41-s + 13·43-s + 2·47-s + 49-s − 53-s − 11·59-s − 13·61-s − 8·71-s − 8·73-s + 4·77-s + 4·79-s + 7·83-s + 14·89-s − 3·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.832·13-s − 1.69·17-s + 1.37·19-s − 1.87·23-s + 0.557·29-s − 1.25·31-s − 1.64·37-s + 0.156·41-s + 1.98·43-s + 0.291·47-s + 1/7·49-s − 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s − 0.936·73-s + 0.455·77-s + 0.450·79-s + 0.768·83-s + 1.48·89-s − 0.314·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.453972908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453972908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.80773289075842, −13.48332349789322, −12.57959055195330, −12.13298058253019, −11.96725307524663, −11.35858802430923, −10.74478331185384, −10.45294089831171, −9.657664102146514, −9.148400350707488, −9.046397664700907, −8.270422370832914, −7.575558903270384, −7.325347518768934, −6.701630492267040, −6.084523207002329, −5.693987803591972, −4.883578724871774, −4.451688568996837, −3.943786641295012, −3.324022348869749, −2.529710621162923, −1.896904444204739, −1.421172567101665, −0.3678858532452510,
0.3678858532452510, 1.421172567101665, 1.896904444204739, 2.529710621162923, 3.324022348869749, 3.943786641295012, 4.451688568996837, 4.883578724871774, 5.693987803591972, 6.084523207002329, 6.701630492267040, 7.325347518768934, 7.575558903270384, 8.270422370832914, 9.046397664700907, 9.148400350707488, 9.657664102146514, 10.45294089831171, 10.74478331185384, 11.35858802430923, 11.96725307524663, 12.13298058253019, 12.57959055195330, 13.48332349789322, 13.80773289075842
