| L(s) = 1 | − 3·2-s − 5·3-s + 4-s − 12·5-s + 15·6-s + 11·7-s + 21·8-s − 2·9-s + 36·10-s − 54·11-s − 5·12-s + 11·13-s − 33·14-s + 60·15-s − 71·16-s − 93·17-s + 6·18-s + 19·19-s − 12·20-s − 55·21-s + 162·22-s + 183·23-s − 105·24-s + 19·25-s − 33·26-s + 145·27-s + 11·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.962·3-s + 1/8·4-s − 1.07·5-s + 1.02·6-s + 0.593·7-s + 0.928·8-s − 0.0740·9-s + 1.13·10-s − 1.48·11-s − 0.120·12-s + 0.234·13-s − 0.629·14-s + 1.03·15-s − 1.10·16-s − 1.32·17-s + 0.0785·18-s + 0.229·19-s − 0.134·20-s − 0.571·21-s + 1.56·22-s + 1.65·23-s − 0.893·24-s + 0.151·25-s − 0.248·26-s + 1.03·27-s + 0.0742·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 11 T + p^{3} T^{2} \) |
| 17 | \( 1 + 93 T + p^{3} T^{2} \) |
| 23 | \( 1 - 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 249 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 250 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 196 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 435 T + p^{3} T^{2} \) |
| 59 | \( 1 - 195 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 + 961 T + p^{3} T^{2} \) |
| 71 | \( 1 + 246 T + p^{3} T^{2} \) |
| 73 | \( 1 - 353 T + p^{3} T^{2} \) |
| 79 | \( 1 + 34 T + p^{3} T^{2} \) |
| 83 | \( 1 - 234 T + p^{3} T^{2} \) |
| 89 | \( 1 + 168 T + p^{3} T^{2} \) |
| 97 | \( 1 - 758 T + p^{3} T^{2} \) |
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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
−17.58875749031053232571110216987, −16.44460263590893958759911056138, −15.31887690195496920901368861056, −13.18979549337806002217766502698, −11.38639438643715075444238680673, −10.72499211422131494197200210212, −8.681385456418463477585042604079, −7.43877395472707668026159534233, −4.94183518858343165992778413003, 0,
4.94183518858343165992778413003, 7.43877395472707668026159534233, 8.681385456418463477585042604079, 10.72499211422131494197200210212, 11.38639438643715075444238680673, 13.18979549337806002217766502698, 15.31887690195496920901368861056, 16.44460263590893958759911056138, 17.58875749031053232571110216987
