| L(s) = 1 | − 2-s − 1.20·3-s + 4-s − 2.40·5-s + 1.20·6-s + 1.52·7-s − 8-s − 1.53·9-s + 2.40·10-s + 5.07·11-s − 1.20·12-s + 5.11·13-s − 1.52·14-s + 2.90·15-s + 16-s − 6.65·17-s + 1.53·18-s − 5.71·19-s − 2.40·20-s − 1.83·21-s − 5.07·22-s − 6.36·23-s + 1.20·24-s + 0.775·25-s − 5.11·26-s + 5.48·27-s + 1.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.698·3-s + 0.5·4-s − 1.07·5-s + 0.493·6-s + 0.574·7-s − 0.353·8-s − 0.512·9-s + 0.759·10-s + 1.53·11-s − 0.349·12-s + 1.41·13-s − 0.406·14-s + 0.750·15-s + 0.250·16-s − 1.61·17-s + 0.362·18-s − 1.31·19-s − 0.537·20-s − 0.401·21-s − 1.08·22-s − 1.32·23-s + 0.246·24-s + 0.155·25-s − 1.00·26-s + 1.05·27-s + 0.287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 + 6.36T + 23T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 - 9.69T + 41T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 + 9.42T + 53T^{2} \) |
| 59 | \( 1 + 6.31T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 2.54T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 + 0.689T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−10.79625730401112495255288429623, −9.295773356753417131919694234028, −8.525830032222663084453420913201, −7.968556319775287584944784928019, −6.50590437742635495620363041277, −6.22355367298792117880628564973, −4.51149399269630591623552569426, −3.69649642647639065917736726946, −1.73496624976198570446485413223, 0,
1.73496624976198570446485413223, 3.69649642647639065917736726946, 4.51149399269630591623552569426, 6.22355367298792117880628564973, 6.50590437742635495620363041277, 7.968556319775287584944784928019, 8.525830032222663084453420913201, 9.295773356753417131919694234028, 10.79625730401112495255288429623
