L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s + 2·13-s − 2·19-s − 2·21-s + 27-s + 8·31-s − 33-s + 2·37-s + 2·39-s + 2·43-s − 3·49-s + 6·53-s − 2·57-s + 12·59-s − 2·61-s − 2·63-s − 4·67-s − 2·73-s + 2·77-s − 10·79-s + 81-s − 12·83-s − 6·89-s − 4·91-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.458·19-s − 0.436·21-s + 0.192·27-s + 1.43·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s + 0.304·43-s − 3/7·49-s + 0.824·53-s − 0.264·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.488·67-s − 0.234·73-s + 0.227·77-s − 1.12·79-s + 1/9·81-s − 1.31·83-s − 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430628835\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430628835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.48863796879474, −13.81944480438467, −13.51043697924065, −12.86527452037489, −12.71617902686244, −11.84040422282253, −11.50815777147166, −10.68434326944505, −10.32821567557825, −9.737136927272812, −9.353742434948307, −8.568175646458011, −8.344660988050072, −7.720084798508408, −6.896806114564162, −6.704782964763217, −5.864194447984705, −5.497543043600164, −4.427205246457396, −4.216071156479047, −3.318996587254597, −2.887861004569329, −2.250303796693778, −1.386274841557224, −0.5340735788272583,
0.5340735788272583, 1.386274841557224, 2.250303796693778, 2.887861004569329, 3.318996587254597, 4.216071156479047, 4.427205246457396, 5.497543043600164, 5.864194447984705, 6.704782964763217, 6.896806114564162, 7.720084798508408, 8.344660988050072, 8.568175646458011, 9.353742434948307, 9.737136927272812, 10.32821567557825, 10.68434326944505, 11.50815777147166, 11.84040422282253, 12.71617902686244, 12.86527452037489, 13.51043697924065, 13.81944480438467, 14.48863796879474