| L(s) = 1 | + 3·3-s − 3·5-s − 3·7-s + 6·9-s + 6·11-s − 9·13-s − 9·15-s − 6·19-s − 9·21-s − 3·23-s + 9·25-s + 10·27-s − 6·29-s + 18·33-s + 9·35-s − 27·39-s + 12·41-s + 9·43-s − 18·45-s + 6·49-s − 9·53-s − 18·55-s − 18·57-s + 27·59-s − 33·61-s − 18·63-s + 27·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s − 1.13·7-s + 2·9-s + 1.80·11-s − 2.49·13-s − 2.32·15-s − 1.37·19-s − 1.96·21-s − 0.625·23-s + 9/5·25-s + 1.92·27-s − 1.11·29-s + 3.13·33-s + 1.52·35-s − 4.32·39-s + 1.87·41-s + 1.37·43-s − 2.68·45-s + 6/7·49-s − 1.23·53-s − 2.42·55-s − 2.38·57-s + 3.51·59-s − 4.22·61-s − 2.26·63-s + 3.34·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.559311222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.559311222\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 3 T - 16 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 48 T^{3} + 18 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 9 T + 30 T^{2} + 70 T^{3} + 30 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 42 T^{2} + 4 T^{3} + 42 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 222 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 86 T^{3} + 18 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 54 T^{2} - 74 T^{3} + 54 p T^{4} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 72 T^{2} + 2 p T^{3} + 72 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 966 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 558 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 69 T^{2} - 232 T^{3} + 69 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 174 T^{2} + 952 T^{3} + 174 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 27 T + 408 T^{2} - 3814 T^{3} + 408 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 33 T + 540 T^{2} + 5288 T^{3} + 540 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 15 T + 264 T^{2} + 2082 T^{3} + 264 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 60 T^{2} - 402 T^{3} + 60 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 18 T + 234 T^{2} - 1942 T^{3} + 234 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 144 T^{2} + 344 T^{3} + 144 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 270 T^{2} - 1910 T^{3} + 270 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 108 T^{2} - 510 T^{3} + 108 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 84 T^{2} + 492 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.14007549500221363358999757182, −6.82096482588298673040405124894, −6.56424004893800812947655026009, −6.36971393201773100983475775651, −6.12477993822057657532343594795, −5.86061346501011050985462171864, −5.64957256346011092664150617399, −5.15843055996828999182670769973, −4.87572433866845635562140879823, −4.56213073495298697879236417317, −4.32392823552072945706795611372, −4.28674609710486782548160537392, −4.18344328181042045884036831113, −3.66844199923267200525129268530, −3.44637116159750082615850509267, −3.37604965244261117652366746810, −2.95176216575052108638916451749, −2.81030178753131464128305461195, −2.46295618529030660385169932401, −2.02508263472788624612758525493, −2.01389359606571696671366565934, −1.70656951286178143510545561438, −0.962137200052548421357751631647, −0.67555425469213376190742891100, −0.32420786596393055999396332525,
0.32420786596393055999396332525, 0.67555425469213376190742891100, 0.962137200052548421357751631647, 1.70656951286178143510545561438, 2.01389359606571696671366565934, 2.02508263472788624612758525493, 2.46295618529030660385169932401, 2.81030178753131464128305461195, 2.95176216575052108638916451749, 3.37604965244261117652366746810, 3.44637116159750082615850509267, 3.66844199923267200525129268530, 4.18344328181042045884036831113, 4.28674609710486782548160537392, 4.32392823552072945706795611372, 4.56213073495298697879236417317, 4.87572433866845635562140879823, 5.15843055996828999182670769973, 5.64957256346011092664150617399, 5.86061346501011050985462171864, 6.12477993822057657532343594795, 6.36971393201773100983475775651, 6.56424004893800812947655026009, 6.82096482588298673040405124894, 7.14007549500221363358999757182
