| L(s) = 1 | + 2·3-s − 5-s + 7·7-s − 9-s + 3·13-s − 2·15-s − 4·17-s + 8·19-s + 14·21-s + 9·23-s − 9·25-s − 7·27-s − 9·29-s + 17·31-s − 7·35-s − 3·37-s + 6·39-s − 16·41-s − 4·43-s + 45-s + 11·47-s + 18·49-s − 8·51-s − 3·53-s + 16·57-s − 2·59-s + 15·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 2.64·7-s − 1/3·9-s + 0.832·13-s − 0.516·15-s − 0.970·17-s + 1.83·19-s + 3.05·21-s + 1.87·23-s − 9/5·25-s − 1.34·27-s − 1.67·29-s + 3.05·31-s − 1.18·35-s − 0.493·37-s + 0.960·39-s − 2.49·41-s − 0.609·43-s + 0.149·45-s + 1.60·47-s + 18/7·49-s − 1.12·51-s − 0.412·53-s + 2.11·57-s − 0.260·59-s + 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25934336 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25934336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.258191018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.258191018\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 2 p T^{2} + 12 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - p T + 31 T^{2} - 94 T^{3} + 31 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T^{2} + 27 T^{3} - 3 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 6 T^{2} - 16 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 31 T^{2} + 120 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 240 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 428 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 9 T + 110 T^{2} + 536 T^{3} + 110 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 184 T^{2} - 1202 T^{3} + 184 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 193 T^{2} + 1359 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 9 T^{2} + 112 T^{3} + 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 11 T + 131 T^{2} - 1030 T^{3} + 131 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 3 T + 59 T^{2} + 610 T^{3} + 59 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 15 T + 212 T^{2} - 1778 T^{3} + 212 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 22 T^{2} - 274 T^{3} + 22 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 714 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 195 T^{2} + 839 T^{3} + 195 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 218 T^{2} + 126 T^{3} + 218 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 9 T + 173 T^{2} + 1606 T^{3} + 173 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2784 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 47 T^{2} + 256 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
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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
−10.50667674794395328904484860416, −10.24559456423596935575688897165, −9.844634532495756245168498798666, −9.547408148510392708352804041462, −8.869996196977424984266331586994, −8.863899326492502878201953865086, −8.627342280285411058871019505602, −8.293683295342721729660434166900, −7.88637576847425579767411056500, −7.83502155766063885308161734339, −7.46566984707687402433081972410, −7.01388337253448340997932032971, −6.71769660084259130592038541006, −6.07682723621157549923626312270, −5.54008877409685999977956279070, −5.46159198185251203645560997923, −4.86530737922999155740176177980, −4.63674996738275428236758451517, −4.27208244246088495069323457097, −3.56361983012300239336936775696, −3.34243059582778935043951132583, −2.84942306405856913627278075228, −2.14416178081521092823242555565, −1.75209515526012064562747615162, −1.12037709954869759125968838726,
1.12037709954869759125968838726, 1.75209515526012064562747615162, 2.14416178081521092823242555565, 2.84942306405856913627278075228, 3.34243059582778935043951132583, 3.56361983012300239336936775696, 4.27208244246088495069323457097, 4.63674996738275428236758451517, 4.86530737922999155740176177980, 5.46159198185251203645560997923, 5.54008877409685999977956279070, 6.07682723621157549923626312270, 6.71769660084259130592038541006, 7.01388337253448340997932032971, 7.46566984707687402433081972410, 7.83502155766063885308161734339, 7.88637576847425579767411056500, 8.293683295342721729660434166900, 8.627342280285411058871019505602, 8.863899326492502878201953865086, 8.869996196977424984266331586994, 9.547408148510392708352804041462, 9.844634532495756245168498798666, 10.24559456423596935575688897165, 10.50667674794395328904484860416
