| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 3·19-s − 2·21-s + 4·23-s + 25-s − 27-s − 7·31-s − 33-s − 2·35-s − 37-s + 39-s + 4·41-s − 7·43-s − 45-s + 10·47-s − 3·49-s − 51-s + 14·53-s − 55-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.688·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.25·31-s − 0.174·33-s − 0.338·35-s − 0.164·37-s + 0.160·39-s + 0.624·41-s − 1.06·43-s − 0.149·45-s + 1.45·47-s − 3/7·49-s − 0.140·51-s + 1.92·53-s − 0.134·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.29363060146385, −12.67662245667727, −12.17379449870940, −11.68807945907595, −11.59425263177416, −10.90447240674026, −10.54911282039175, −10.10990243744927, −9.422740952218492, −8.980272110875281, −8.592298898434964, −7.900950142291073, −7.419924611638625, −7.171941936993767, −6.598475879461316, −5.904033247090933, −5.343916346172615, −5.144236085896783, −4.419670184485242, −3.990926906818811, −3.409935359852420, −2.749355094319344, −2.048715326648558, −1.342702570551904, −0.8554135348452387, 0,
0.8554135348452387, 1.342702570551904, 2.048715326648558, 2.749355094319344, 3.409935359852420, 3.990926906818811, 4.419670184485242, 5.144236085896783, 5.343916346172615, 5.904033247090933, 6.598475879461316, 7.171941936993767, 7.419924611638625, 7.900950142291073, 8.592298898434964, 8.980272110875281, 9.422740952218492, 10.10990243744927, 10.54911282039175, 10.90447240674026, 11.59425263177416, 11.68807945907595, 12.17379449870940, 12.67662245667727, 13.29363060146385
