| L(s) = 1 | − 2-s − 3·3-s − 7·4-s − 4·5-s + 3·6-s − 26·7-s + 15·8-s + 9·9-s + 4·10-s + 11·11-s + 21·12-s − 32·13-s + 26·14-s + 12·15-s + 41·16-s + 74·17-s − 9·18-s − 60·19-s + 28·20-s + 78·21-s − 11·22-s − 182·23-s − 45·24-s − 109·25-s + 32·26-s − 27·27-s + 182·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.357·5-s + 0.204·6-s − 1.40·7-s + 0.662·8-s + 1/3·9-s + 0.126·10-s + 0.301·11-s + 0.505·12-s − 0.682·13-s + 0.496·14-s + 0.206·15-s + 0.640·16-s + 1.05·17-s − 0.117·18-s − 0.724·19-s + 0.313·20-s + 0.810·21-s − 0.106·22-s − 1.64·23-s − 0.382·24-s − 0.871·25-s + 0.241·26-s − 0.192·27-s + 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 182 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 66 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 - 408 T + p^{3} T^{2} \) |
| 47 | \( 1 + 506 T + p^{3} T^{2} \) |
| 53 | \( 1 - 348 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 132 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 762 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 + 550 T + p^{3} T^{2} \) |
| 83 | \( 1 + 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 570 T + p^{3} T^{2} \) |
| 97 | \( 1 - 14 T + p^{3} T^{2} \) |
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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
−15.99800527471375409681628617840, −14.38916713662311871880750714929, −13.00270062249319427861056346089, −12.08948336874890052910498086982, −10.24583505954726159699044737823, −9.411879972450193940857107479417, −7.67922620982539398739813089313, −5.93154960330735044974422265346, −3.99071285874632850409323170230, 0,
3.99071285874632850409323170230, 5.93154960330735044974422265346, 7.67922620982539398739813089313, 9.411879972450193940857107479417, 10.24583505954726159699044737823, 12.08948336874890052910498086982, 13.00270062249319427861056346089, 14.38916713662311871880750714929, 15.99800527471375409681628617840
