| L(s) = 1 | − 8·7-s − 8·11-s − 48·13-s − 8·17-s − 72·23-s + 44·31-s − 112·37-s − 32·41-s − 104·43-s − 80·47-s + 32·49-s − 104·53-s − 180·61-s − 264·67-s + 256·71-s − 112·73-s + 64·77-s − 9·81-s + 16·83-s + 384·91-s − 320·97-s − 496·101-s − 144·103-s + 264·107-s − 32·113-s + 64·119-s − 396·121-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 0.727·11-s − 3.69·13-s − 0.470·17-s − 3.13·23-s + 1.41·31-s − 3.02·37-s − 0.780·41-s − 2.41·43-s − 1.70·47-s + 0.653·49-s − 1.96·53-s − 2.95·61-s − 3.94·67-s + 3.60·71-s − 1.53·73-s + 0.831·77-s − 1/9·81-s + 0.192·83-s + 4.21·91-s − 3.29·97-s − 4.91·101-s − 1.39·103-s + 2.46·107-s − 0.283·113-s + 0.537·119-s − 3.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001305857126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001305857126\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 432 T^{3} + 5807 T^{4} + 432 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 222 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 18336 T^{3} + 246479 T^{4} + 18336 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 2280 T^{3} + 162434 T^{4} + 2280 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1346 T^{2} + 711171 T^{4} - 1346 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 72 T + 2592 T^{2} + 76968 T^{3} + 1993922 T^{4} + 76968 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1004 T^{2} + 1647750 T^{4} - 1004 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 22 T + 1827 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 280560 T^{3} + 11259554 T^{4} + 280560 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 3402 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 104 T + 5408 T^{2} + 220272 T^{3} + 8899487 T^{4} + 220272 p^{2} T^{5} + 5408 p^{4} T^{6} + 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 80 T + 3200 T^{2} + 193680 T^{3} + 11677538 T^{4} + 193680 p^{2} T^{5} + 3200 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 104 T + 5408 T^{2} + 113256 T^{3} - 586558 T^{4} + 113256 p^{2} T^{5} + 5408 p^{4} T^{6} + 104 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1628 T^{2} + 24799014 T^{4} - 1628 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 90 T + 6011 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 264 T + 34848 T^{2} + 3306864 T^{3} + 249207983 T^{4} + 3306864 p^{2} T^{5} + 34848 p^{4} T^{6} + 264 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 128 T + 13002 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 638064 T^{3} + 64776194 T^{4} + 638064 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12604 T^{2} + 115279110 T^{4} - 12604 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} + 124464 T^{3} - 94124542 T^{4} + 124464 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 20708 T^{2} + 217938054 T^{4} - 20708 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 320 T + 51200 T^{2} + 6683520 T^{3} + 740728463 T^{4} + 6683520 p^{2} T^{5} + 51200 p^{4} T^{6} + 320 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33241080732849044633232988213, −7.20845326975153368204050146772, −7.16151654352557774050383410166, −6.52257882455891355052298728287, −6.50384797373009716465553482795, −6.41229432760245449821795599005, −6.37653707416098295449287816465, −5.44957059220188541342801673427, −5.41467862237265875654455921221, −5.39774725045213043516975291483, −5.20129081631529646454295029516, −4.60112257391843491453899437262, −4.42899950434013199452085782468, −4.37729282718767878483079687374, −4.11460403268952339938733937465, −3.45994880022650168411049572303, −3.12207059081135935310591632156, −3.00205359817033173985379758877, −2.90883034803527996528012969083, −2.43272682974622000599008853865, −1.88979607075211400477369658870, −1.72167649489239662330427106253, −1.63963740417030090290695992691, −0.16229429237206093483946942625, −0.02770792013555337644591487522,
0.02770792013555337644591487522, 0.16229429237206093483946942625, 1.63963740417030090290695992691, 1.72167649489239662330427106253, 1.88979607075211400477369658870, 2.43272682974622000599008853865, 2.90883034803527996528012969083, 3.00205359817033173985379758877, 3.12207059081135935310591632156, 3.45994880022650168411049572303, 4.11460403268952339938733937465, 4.37729282718767878483079687374, 4.42899950434013199452085782468, 4.60112257391843491453899437262, 5.20129081631529646454295029516, 5.39774725045213043516975291483, 5.41467862237265875654455921221, 5.44957059220188541342801673427, 6.37653707416098295449287816465, 6.41229432760245449821795599005, 6.50384797373009716465553482795, 6.52257882455891355052298728287, 7.16151654352557774050383410166, 7.20845326975153368204050146772, 7.33241080732849044633232988213
