| L(s) = 1 | − 4.42e3·5-s + 6.71e3·9-s − 4.78e3·13-s + 4.64e5·17-s + 1.07e7·25-s − 6.06e6·29-s + 5.96e5·37-s − 4.92e7·41-s − 2.96e7·45-s + 5.77e5·49-s − 2.07e8·53-s − 1.87e8·61-s + 2.11e7·65-s + 6.89e8·73-s − 3.42e8·81-s − 2.05e9·85-s − 2.51e8·89-s − 3.92e8·97-s + 9.78e8·101-s − 1.94e8·109-s − 2.16e9·113-s − 3.21e7·117-s + 5.79e8·121-s − 1.72e10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.16·5-s + 0.341·9-s − 0.0464·13-s + 1.34·17-s + 5.50·25-s − 1.59·29-s + 0.0522·37-s − 2.72·41-s − 1.07·45-s + 0.0143·49-s − 3.60·53-s − 1.73·61-s + 0.147·65-s + 2.84·73-s − 0.883·81-s − 4.26·85-s − 0.424·89-s − 0.449·97-s + 0.935·101-s − 0.132·109-s − 1.25·113-s − 0.0158·117-s + 0.245·121-s − 6.32·125-s + 5.03·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1698956568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1698956568\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 746 p^{2} T^{2} + p^{18} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 442 p T + p^{9} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 11794 p^{2} T^{2} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 579664298 T^{2} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2394 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 13666 p T + p^{9} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 581972587238 T^{2} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2978949644206 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3030250 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 46056629199422 T^{2} + p^{18} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 298078 T + p^{9} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 24607670 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 48674756673686 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2004687397436254 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 103629586 T + p^{9} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14209809408271798 T^{2} + p^{18} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 93965290 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 28747478586246586 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 86686368197120782 T^{2} + p^{18} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 344846394 T + p^{9} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 223919874811772318 T^{2} + p^{18} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 143547174371031194 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 125586230 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 196001262 T + p^{9} T^{2} )^{2} \) |
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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
−15.35575243347091694031585735708, −14.71623218099739080931282127411, −14.02945192860150036782402132932, −12.96923830812526089656773067112, −12.22686062401685545524290313390, −12.17019506509891214010471536929, −11.22140938213930204312159780310, −11.13578442191669006717982735340, −10.07681846210321433482555836739, −9.160873049967083513844687948207, −8.157391138479192441532964494148, −7.890746801017260082356707444032, −7.39624768297040978857607770776, −6.58962508807937463697468517926, −5.14657188753370993099087096872, −4.42214410471668150992441993261, −3.48922628133317646791160430582, −3.36423097397502554959584464241, −1.39036584303289819483461346411, −0.17296433096695546437435193014,
0.17296433096695546437435193014, 1.39036584303289819483461346411, 3.36423097397502554959584464241, 3.48922628133317646791160430582, 4.42214410471668150992441993261, 5.14657188753370993099087096872, 6.58962508807937463697468517926, 7.39624768297040978857607770776, 7.890746801017260082356707444032, 8.157391138479192441532964494148, 9.160873049967083513844687948207, 10.07681846210321433482555836739, 11.13578442191669006717982735340, 11.22140938213930204312159780310, 12.17019506509891214010471536929, 12.22686062401685545524290313390, 12.96923830812526089656773067112, 14.02945192860150036782402132932, 14.71623218099739080931282127411, 15.35575243347091694031585735708
