| L(s) = 1 | + 2·2-s + 4·4-s − 9·5-s − 31·7-s + 8·8-s − 18·10-s − 15·11-s − 37·13-s − 62·14-s + 16·16-s − 42·17-s − 28·19-s − 36·20-s − 30·22-s + 195·23-s − 44·25-s − 74·26-s − 124·28-s + 111·29-s − 205·31-s + 32·32-s − 84·34-s + 279·35-s − 166·37-s − 56·38-s − 72·40-s − 261·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.804·5-s − 1.67·7-s + 0.353·8-s − 0.569·10-s − 0.411·11-s − 0.789·13-s − 1.18·14-s + 1/4·16-s − 0.599·17-s − 0.338·19-s − 0.402·20-s − 0.290·22-s + 1.76·23-s − 0.351·25-s − 0.558·26-s − 0.836·28-s + 0.710·29-s − 1.18·31-s + 0.176·32-s − 0.423·34-s + 1.34·35-s − 0.737·37-s − 0.239·38-s − 0.284·40-s − 0.994·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 31 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 37 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 195 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 205 T + p^{3} T^{2} \) |
| 37 | \( 1 + 166 T + p^{3} T^{2} \) |
| 41 | \( 1 + 261 T + p^{3} T^{2} \) |
| 43 | \( 1 + p T + p^{3} T^{2} \) |
| 47 | \( 1 - 177 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 - 159 T + p^{3} T^{2} \) |
| 61 | \( 1 - 191 T + p^{3} T^{2} \) |
| 67 | \( 1 + 421 T + p^{3} T^{2} \) |
| 71 | \( 1 - 156 T + p^{3} T^{2} \) |
| 73 | \( 1 - 182 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1133 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1083 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1050 T + p^{3} T^{2} \) |
| 97 | \( 1 + 901 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19716085641036855447796100935, −11.03888068542885145618094628924, −10.00408977443763000520444292620, −8.838873136956922241005569597564, −7.32640427102628944943240448901, −6.60776506560218590184122898675, −5.18741973227182015123376982753, −3.81606404181570300240261140799, −2.76392463632484514857915266629, 0,
2.76392463632484514857915266629, 3.81606404181570300240261140799, 5.18741973227182015123376982753, 6.60776506560218590184122898675, 7.32640427102628944943240448901, 8.838873136956922241005569597564, 10.00408977443763000520444292620, 11.03888068542885145618094628924, 12.19716085641036855447796100935
