| L(s) = 1 | − 3-s + 4·7-s + 9-s + 11-s + 13-s + 4·19-s − 4·21-s − 5·25-s − 27-s + 10·31-s − 33-s + 2·37-s − 39-s − 6·41-s + 10·43-s + 9·49-s + 6·53-s − 4·57-s + 2·61-s + 4·63-s − 2·67-s − 12·71-s − 10·73-s + 5·75-s + 4·77-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.917·19-s − 0.872·21-s − 25-s − 0.192·27-s + 1.79·31-s − 0.174·33-s + 0.328·37-s − 0.160·39-s − 0.937·41-s + 1.52·43-s + 9/7·49-s + 0.824·53-s − 0.529·57-s + 0.256·61-s + 0.503·63-s − 0.244·67-s − 1.42·71-s − 1.17·73-s + 0.577·75-s + 0.455·77-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273307491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273307491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.84597132115712663899559375880, −7.42637273851055462147132262146, −6.46762854824131194502412119589, −5.80399630697470544966076570139, −5.10502715973708141214440798976, −4.49716507625037401416860734725, −3.79051690554858010200238005157, −2.61199029985494601733520721479, −1.62020718141949052454531673851, −0.864710413570456851270462671097,
0.864710413570456851270462671097, 1.62020718141949052454531673851, 2.61199029985494601733520721479, 3.79051690554858010200238005157, 4.49716507625037401416860734725, 5.10502715973708141214440798976, 5.80399630697470544966076570139, 6.46762854824131194502412119589, 7.42637273851055462147132262146, 7.84597132115712663899559375880
