L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s − 5-s + 9·6-s + 3·8-s + 6·9-s − 3·10-s + 6·11-s + 12·12-s − 3·15-s + 3·16-s + 6·17-s + 18·18-s − 6·19-s − 4·20-s + 18·22-s + 6·23-s + 9·24-s + 9·27-s − 9·30-s − 6·31-s + 6·32-s + 18·33-s + 18·34-s + 24·36-s + 2·37-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s − 0.447·5-s + 3.67·6-s + 1.06·8-s + 2·9-s − 0.948·10-s + 1.80·11-s + 3.46·12-s − 0.774·15-s + 3/4·16-s + 1.45·17-s + 4.24·18-s − 1.37·19-s − 0.894·20-s + 3.83·22-s + 1.25·23-s + 1.83·24-s + 1.73·27-s − 1.64·30-s − 1.07·31-s + 1.06·32-s + 3.13·33-s + 3.08·34-s + 4·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.67017317\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.67017317\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36635298589507719654214645768, −10.36236119903694394554323921640, −9.599927542121896174984728151404, −9.345971774434434575531338480365, −8.707195568926594682669098306317, −8.520801828722537371656726492452, −7.963242716891906863961057124400, −7.57981962027870146132822947943, −6.86514974907766630742974976438, −6.67074874451687845803040854723, −6.20992263012454982622664727193, −5.41456778976189495253697094747, −4.92895676425564098916997696251, −4.59444308729533145261804357474, −3.89357039044115016319593581900, −3.71208336119656193778061029452, −3.26997031979120518152952735536, −2.98441676926231538447324610862, −1.83216051102864125002281528725, −1.47208837573094888561336859501,
1.47208837573094888561336859501, 1.83216051102864125002281528725, 2.98441676926231538447324610862, 3.26997031979120518152952735536, 3.71208336119656193778061029452, 3.89357039044115016319593581900, 4.59444308729533145261804357474, 4.92895676425564098916997696251, 5.41456778976189495253697094747, 6.20992263012454982622664727193, 6.67074874451687845803040854723, 6.86514974907766630742974976438, 7.57981962027870146132822947943, 7.963242716891906863961057124400, 8.520801828722537371656726492452, 8.707195568926594682669098306317, 9.345971774434434575531338480365, 9.599927542121896174984728151404, 10.36236119903694394554323921640, 10.36635298589507719654214645768