| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s − 21-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 2·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.92689550954325, −14.36824007078907, −13.88351143902959, −13.31658213074651, −12.64344487709926, −12.22437988182550, −11.62676216467067, −11.26606632380116, −10.52608620096870, −10.34480091579109, −9.618272642679951, −9.114867985622957, −8.749937674872430, −7.854765757859092, −7.416003157985926, −7.078612919289955, −6.089976221807357, −5.922522892352376, −5.168689039366721, −4.650999016659579, −3.788881501719787, −3.141174182217834, −2.162523232055773, −1.756769089317631, −0.8953739495449851, 0,
0.8953739495449851, 1.756769089317631, 2.162523232055773, 3.141174182217834, 3.788881501719787, 4.650999016659579, 5.168689039366721, 5.922522892352376, 6.089976221807357, 7.078612919289955, 7.416003157985926, 7.854765757859092, 8.749937674872430, 9.114867985622957, 9.618272642679951, 10.34480091579109, 10.52608620096870, 11.26606632380116, 11.62676216467067, 12.22437988182550, 12.64344487709926, 13.31658213074651, 13.88351143902959, 14.36824007078907, 14.92689550954325
