| L(s) = 1 | + 2-s − 2.63·3-s + 4-s − 3.59·5-s − 2.63·6-s − 1.69·7-s + 8-s + 3.95·9-s − 3.59·10-s − 11-s − 2.63·12-s − 0.973·13-s − 1.69·14-s + 9.47·15-s + 16-s − 2.03·17-s + 3.95·18-s + 2.88·19-s − 3.59·20-s + 4.45·21-s − 22-s + 6.67·23-s − 2.63·24-s + 7.92·25-s − 0.973·26-s − 2.51·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s − 1.60·5-s − 1.07·6-s − 0.639·7-s + 0.353·8-s + 1.31·9-s − 1.13·10-s − 0.301·11-s − 0.761·12-s − 0.269·13-s − 0.451·14-s + 2.44·15-s + 0.250·16-s − 0.493·17-s + 0.931·18-s + 0.662·19-s − 0.803·20-s + 0.972·21-s − 0.213·22-s + 1.39·23-s − 0.538·24-s + 1.58·25-s − 0.190·26-s − 0.483·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 2.63T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 13 | \( 1 + 0.973T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 - 6.67T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 0.490T + 43T^{2} \) |
| 47 | \( 1 + 1.95T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 7.69T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−7.57292523907777317488005830777, −7.18720998990210294864906130735, −6.47473846631273836477030354124, −5.76317468946614210340870808172, −4.85051667473603187525651169424, −4.52517979498326030068675363135, −3.54008069269570275456285363633, −2.79998399881144503886139375095, −1.00211684270100190405135626443, 0,
1.00211684270100190405135626443, 2.79998399881144503886139375095, 3.54008069269570275456285363633, 4.52517979498326030068675363135, 4.85051667473603187525651169424, 5.76317468946614210340870808172, 6.47473846631273836477030354124, 7.18720998990210294864906130735, 7.57292523907777317488005830777
