| L(s) = 1 | − 2·2-s − 4·4-s + 5·5-s + 24·8-s − 10·10-s + 10·11-s − 80·13-s − 16·16-s + 7·17-s − 113·19-s − 20·20-s − 20·22-s − 81·23-s + 25·25-s + 160·26-s − 220·29-s − 189·31-s − 160·32-s − 14·34-s + 170·37-s + 226·38-s + 120·40-s − 130·41-s + 10·43-s − 40·44-s + 162·46-s + 160·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.274·11-s − 1.70·13-s − 1/4·16-s + 0.0998·17-s − 1.36·19-s − 0.223·20-s − 0.193·22-s − 0.734·23-s + 1/5·25-s + 1.20·26-s − 1.40·29-s − 1.09·31-s − 0.883·32-s − 0.0706·34-s + 0.755·37-s + 0.964·38-s + 0.474·40-s − 0.495·41-s + 0.0354·43-s − 0.137·44-s + 0.519·46-s + 0.496·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 + 80 T + p^{3} T^{2} \) |
| 17 | \( 1 - 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 113 T + p^{3} T^{2} \) |
| 23 | \( 1 + 81 T + p^{3} T^{2} \) |
| 29 | \( 1 + 220 T + p^{3} T^{2} \) |
| 31 | \( 1 + 189 T + p^{3} T^{2} \) |
| 37 | \( 1 - 170 T + p^{3} T^{2} \) |
| 41 | \( 1 + 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 10 T + p^{3} T^{2} \) |
| 47 | \( 1 - 160 T + p^{3} T^{2} \) |
| 53 | \( 1 - 631 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 229 T + p^{3} T^{2} \) |
| 67 | \( 1 - 750 T + p^{3} T^{2} \) |
| 71 | \( 1 - 890 T + p^{3} T^{2} \) |
| 73 | \( 1 + 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 27 T + p^{3} T^{2} \) |
| 83 | \( 1 - 429 T + p^{3} T^{2} \) |
| 89 | \( 1 + 750 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1480 T + p^{3} T^{2} \) |
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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
−12.36551583347829401166892986013, −10.93892655889584767736011162374, −9.906399364360154831934389757019, −9.272955834890351585263412951435, −8.108507795497751933118367780630, −7.03266955627461640977804320995, −5.44862821419945983963792486180, −4.18628886978418125446402177478, −2.04004367435515896269431503176, 0,
2.04004367435515896269431503176, 4.18628886978418125446402177478, 5.44862821419945983963792486180, 7.03266955627461640977804320995, 8.108507795497751933118367780630, 9.272955834890351585263412951435, 9.906399364360154831934389757019, 10.93892655889584767736011162374, 12.36551583347829401166892986013
