| L(s) = 1 | + 3·5-s − 5·7-s + 2·11-s + 4·13-s − 2·17-s − 4·19-s − 3·23-s + 6·25-s − 7·29-s − 8·31-s − 15·35-s + 6·37-s − 13·41-s − 10·43-s − 13·47-s + 49-s − 2·53-s + 6·55-s + 2·59-s + 61-s + 12·65-s − 11·67-s + 10·71-s − 8·73-s − 10·77-s − 2·79-s + 15·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.88·7-s + 0.603·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.625·23-s + 6/5·25-s − 1.29·29-s − 1.43·31-s − 2.53·35-s + 0.986·37-s − 2.03·41-s − 1.52·43-s − 1.89·47-s + 1/7·49-s − 0.274·53-s + 0.809·55-s + 0.260·59-s + 0.128·61-s + 1.48·65-s − 1.34·67-s + 1.18·71-s − 0.936·73-s − 1.13·77-s − 0.225·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T + 24 T^{2} + 67 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 32 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 100 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 56 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 36 T^{2} + 21 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 2 p T^{2} + 355 T^{3} + 2 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 33 T^{2} + 28 T^{3} + 33 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 99 T^{2} - 440 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 142 T^{2} + 1069 T^{3} + 142 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 856 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 13 T + 130 T^{2} + 853 T^{3} + 130 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2 T + 139 T^{2} + 188 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 157 T^{2} - 212 T^{3} + 157 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 146 T^{2} - 193 T^{3} + 146 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 11 T + 162 T^{2} + 967 T^{3} + 162 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 121 T^{2} - 712 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 155 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 153 T^{2} + 340 T^{3} + 153 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 15 T + 276 T^{2} - 2409 T^{3} + 276 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 + 18 T + 255 T^{2} + 2188 T^{3} + 255 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60766967306413998079362742041, −6.86822794790758268308604090603, −6.85945367866195320532742699818, −6.66278354336874147012544659793, −6.36551637653939071979592680707, −6.34467336163841546752847952626, −6.25220282878103190171832554402, −5.80814902969322553205423128040, −5.53941954716165202450591291201, −5.49813949181671930612025771738, −4.98166640668142086956728166208, −4.85049019855600148771862771948, −4.69018578370670787561476512557, −3.99039131171083223824875772646, −3.97628800370832098310534042979, −3.77584802795874328197499038334, −3.32150601864616973102661349024, −3.27375094941645161405477952772, −3.09431314429270189602011295914, −2.47713724730211140635973892664, −2.26846248514422335763099971850, −2.09673202264185550841939696409, −1.51791294628394743942830573327, −1.37310569379430585459189333856, −1.25792480876644243303975716140, 0, 0, 0,
1.25792480876644243303975716140, 1.37310569379430585459189333856, 1.51791294628394743942830573327, 2.09673202264185550841939696409, 2.26846248514422335763099971850, 2.47713724730211140635973892664, 3.09431314429270189602011295914, 3.27375094941645161405477952772, 3.32150601864616973102661349024, 3.77584802795874328197499038334, 3.97628800370832098310534042979, 3.99039131171083223824875772646, 4.69018578370670787561476512557, 4.85049019855600148771862771948, 4.98166640668142086956728166208, 5.49813949181671930612025771738, 5.53941954716165202450591291201, 5.80814902969322553205423128040, 6.25220282878103190171832554402, 6.34467336163841546752847952626, 6.36551637653939071979592680707, 6.66278354336874147012544659793, 6.85945367866195320532742699818, 6.86822794790758268308604090603, 7.60766967306413998079362742041
