| L(s) = 1 | + 10·5-s − 10·7-s + 18·11-s − 34·13-s − 48·17-s + 56·19-s + 30·23-s + 25·25-s − 42·29-s − 256·31-s − 100·35-s + 296·37-s − 222·41-s − 418·43-s − 12·47-s + 90·49-s + 96·53-s + 180·55-s − 354·59-s − 838·61-s − 340·65-s − 1.24e3·67-s − 924·71-s + 1.77e3·73-s − 180·77-s − 1.15e3·79-s + 1.14e3·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.539·7-s + 0.493·11-s − 0.725·13-s − 0.684·17-s + 0.676·19-s + 0.271·23-s + 1/5·25-s − 0.268·29-s − 1.48·31-s − 0.482·35-s + 1.31·37-s − 0.845·41-s − 1.48·43-s − 0.0372·47-s + 0.262·49-s + 0.248·53-s + 0.441·55-s − 0.781·59-s − 1.75·61-s − 0.648·65-s − 2.26·67-s − 1.54·71-s + 2.84·73-s − 0.266·77-s − 1.64·79-s + 1.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1016708579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1016708579\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 10 T + 10 T^{2} - 5960 T^{3} - 143849 T^{4} - 5960 p^{3} T^{5} + 10 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 18 T - 1798 T^{2} + 9720 T^{3} + 2300079 T^{4} + 9720 p^{3} T^{5} - 1798 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 34 T - 2906 T^{2} - 11288 T^{3} + 9133303 T^{4} - 11288 p^{3} T^{5} - 2906 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 24 T + 9349 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 28 T + 13293 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30 T - 13723 T^{2} + 291330 T^{3} + 54845940 T^{4} + 291330 p^{3} T^{5} - 13723 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 42 T - 31930 T^{2} - 633528 T^{3} + 497440119 T^{4} - 633528 p^{3} T^{5} - 31930 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 256 T + 19999 T^{2} - 3595520 T^{3} - 766268288 T^{4} - 3595520 p^{3} T^{5} + 19999 p^{6} T^{6} + 256 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 222 T - 50578 T^{2} - 8431560 T^{3} + 1825612239 T^{4} - 8431560 p^{3} T^{5} - 50578 p^{6} T^{6} + 222 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 418 T - 12446 T^{2} + 11769208 T^{3} + 14819464783 T^{4} + 11769208 p^{3} T^{5} - 12446 p^{6} T^{6} + 418 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T - 145438 T^{2} - 744768 T^{3} + 10399952883 T^{4} - 744768 p^{3} T^{5} - 145438 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 48 T + 118861 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 p T - 286342 T^{2} + 5400 p T^{3} + 101544500559 T^{4} + 5400 p^{4} T^{5} - 286342 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 838 T + 95077 T^{2} + 128385790 T^{3} + 157735346164 T^{4} + 128385790 p^{3} T^{5} + 95077 p^{6} T^{6} + 838 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1240 T + 574030 T^{2} + 448934560 T^{3} + 375555573931 T^{4} + 448934560 p^{3} T^{5} + 574030 p^{6} T^{6} + 1240 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 462 T + 316474 T^{2} + 462 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 886 T + 700422 T^{2} - 886 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1156 T + 21763 T^{2} + 379740220 T^{3} + 826148951704 T^{4} + 379740220 p^{3} T^{5} + 21763 p^{6} T^{6} + 1156 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1146 T + 328277 T^{2} + 181681110 T^{3} - 107575557540 T^{4} + 181681110 p^{3} T^{5} + 328277 p^{6} T^{6} - 1146 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 150 T + 1409974 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 784 T - 1205378 T^{2} - 4164608 T^{3} + 1918571820739 T^{4} - 4164608 p^{3} T^{5} - 1205378 p^{6} T^{6} + 784 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.32834440727651234907312010032, −6.28308730280209992247996623124, −5.78237220079593002885345976732, −5.68083798218779670455200677967, −5.57346628868798261164943666659, −5.25782774775513622777465652611, −4.96000249205269303861416986305, −4.87469662850903091398053194819, −4.48036721781650494404838539015, −4.36168396348757629680304581334, −4.26865044784176658031509555054, −3.80299333410470938163756811699, −3.40096487614082943402396712019, −3.37725204880363657667892370204, −3.23034199951019992710276065845, −2.75902254570154864297032359171, −2.72513435934439476417132851569, −2.19817414366813298802530399118, −2.05416011373715669652022818335, −1.81005599507184486496156518408, −1.49484828778036034335357014782, −1.26342665359376549794664164560, −0.877849678506526531314915767479, −0.45493524988342680123375729753, −0.03927333388567705552260916062,
0.03927333388567705552260916062, 0.45493524988342680123375729753, 0.877849678506526531314915767479, 1.26342665359376549794664164560, 1.49484828778036034335357014782, 1.81005599507184486496156518408, 2.05416011373715669652022818335, 2.19817414366813298802530399118, 2.72513435934439476417132851569, 2.75902254570154864297032359171, 3.23034199951019992710276065845, 3.37725204880363657667892370204, 3.40096487614082943402396712019, 3.80299333410470938163756811699, 4.26865044784176658031509555054, 4.36168396348757629680304581334, 4.48036721781650494404838539015, 4.87469662850903091398053194819, 4.96000249205269303861416986305, 5.25782774775513622777465652611, 5.57346628868798261164943666659, 5.68083798218779670455200677967, 5.78237220079593002885345976732, 6.28308730280209992247996623124, 6.32834440727651234907312010032
