| L(s) = 1 | + 3·2-s + 2·4-s + 12·5-s + 7-s − 3·8-s + 36·10-s − 24·11-s − 11·13-s + 3·14-s − 3·16-s + 70·19-s + 24·20-s − 72·22-s + 12·23-s + 71·25-s − 33·26-s + 2·28-s + 48·29-s + 25·31-s − 12·32-s + 12·35-s + 70·37-s + 210·38-s − 36·40-s − 132·41-s + 37·43-s − 48·44-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 1/2·4-s + 12/5·5-s + 1/7·7-s − 3/8·8-s + 18/5·10-s − 2.18·11-s − 0.846·13-s + 3/14·14-s − 0.187·16-s + 3.68·19-s + 6/5·20-s − 3.27·22-s + 0.521·23-s + 2.83·25-s − 1.26·26-s + 1/14·28-s + 1.65·29-s + 0.806·31-s − 3/8·32-s + 0.342·35-s + 1.89·37-s + 5.52·38-s − 0.899·40-s − 3.21·41-s + 0.860·43-s − 1.09·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.066205813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.066205813\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T - 48 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 24 T + 313 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 11 T - 48 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 577 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 48 T + 1609 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 25 T - 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 132 T + 7489 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 37 T - 480 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 96 T + 5281 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 3529 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 22 T - 4005 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 6937 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 11954 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 9408 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.16720256043048746622969001990, −11.97850395148594652092409009023, −11.30312955681253992571963281928, −10.49542155419057272550633242385, −10.00206032502146487627040238552, −9.993345259607126744209131517834, −9.465923812956025430893007962220, −8.910510287622884519845330055148, −7.921929144781305130377577997000, −7.72940124129270215043332669705, −6.90288776115135209612658406446, −6.29035315922526916260661081823, −5.48742530793269106489739808755, −5.44355733459511986709593205456, −4.89679611519374503348306914215, −4.77942012316488063751754149956, −3.12062856852199609452743453960, −3.02850370797812425897872966736, −2.24106474784551606867477875669, −1.14595377085938302624036586244,
1.14595377085938302624036586244, 2.24106474784551606867477875669, 3.02850370797812425897872966736, 3.12062856852199609452743453960, 4.77942012316488063751754149956, 4.89679611519374503348306914215, 5.44355733459511986709593205456, 5.48742530793269106489739808755, 6.29035315922526916260661081823, 6.90288776115135209612658406446, 7.72940124129270215043332669705, 7.921929144781305130377577997000, 8.910510287622884519845330055148, 9.465923812956025430893007962220, 9.993345259607126744209131517834, 10.00206032502146487627040238552, 10.49542155419057272550633242385, 11.30312955681253992571963281928, 11.97850395148594652092409009023, 12.16720256043048746622969001990
