| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s − 21-s + 4·22-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 28-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 − 1.20·11-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 + 0.852·22-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.06525546003818, −14.25489587643272, −14.10551281604605, −13.03683265805533, −12.69208751410948, −12.28523564077496, −11.68606818546520, −11.06629806193759, −10.70187278099819, −10.35928381735152, −9.722523106804547, −9.038085732499462, −8.542348701014669, −7.917705887898369, −7.456132402703496, −7.151974566452223, −6.175511749199316, −5.840333453673666, −5.129340178744872, −4.556923492598574, −3.890649805225075, −3.095860198326717, −2.283483484770649, −1.767700557008486, −0.6886169165499228, 0,
0.6886169165499228, 1.767700557008486, 2.283483484770649, 3.095860198326717, 3.890649805225075, 4.556923492598574, 5.129340178744872, 5.840333453673666, 6.175511749199316, 7.151974566452223, 7.456132402703496, 7.917705887898369, 8.542348701014669, 9.038085732499462, 9.722523106804547, 10.35928381735152, 10.70187278099819, 11.06629806193759, 11.68606818546520, 12.28523564077496, 12.69208751410948, 13.03683265805533, 14.10551281604605, 14.25489587643272, 15.06525546003818
