| 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 | \( 1 + 901 T + p^{3} T^{2} \) |
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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
−12.00233694210455583132269709395, −10.49084250587472579635532929567, −9.708434387655531672536319104145, −9.235013556988423397630700181999, −7.68497038230648195921060986768, −6.52996595255696752320889082464, −5.73320443153005755121658124085, −3.60114648545183038779602919539, −2.10822466650392394975477454917, 0,
2.10822466650392394975477454917, 3.60114648545183038779602919539, 5.73320443153005755121658124085, 6.52996595255696752320889082464, 7.68497038230648195921060986768, 9.235013556988423397630700181999, 9.708434387655531672536319104145, 10.49084250587472579635532929567, 12.00233694210455583132269709395
