| L(s) = 1 | + 4·2-s + 6·3-s + 4·4-s − 12·5-s + 24·6-s − 16·8-s + 9·9-s − 48·10-s + 4·11-s + 24·12-s + 96·13-s − 72·15-s − 64·16-s − 84·17-s + 36·18-s + 72·19-s − 48·20-s + 16·22-s + 308·23-s − 96·24-s + 188·25-s + 384·26-s − 54·27-s − 160·29-s − 288·30-s − 384·31-s − 64·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.07·5-s + 1.63·6-s − 0.707·8-s + 1/3·9-s − 1.51·10-s + 0.109·11-s + 0.577·12-s + 2.04·13-s − 1.23·15-s − 16-s − 1.19·17-s + 0.471·18-s + 0.869·19-s − 0.536·20-s + 0.155·22-s + 2.79·23-s − 0.816·24-s + 1.50·25-s + 2.89·26-s − 0.384·27-s − 1.02·29-s − 1.75·30-s − 2.22·31-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.013643839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013643839\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T - 44 T^{2} - 744 T^{3} + 1719 T^{4} - 744 p^{3} T^{5} - 44 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 878 T^{2} + 14096 T^{3} - 1049813 T^{4} + 14096 p^{3} T^{5} + 878 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 48 T + 4088 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 84 T - 4436 T^{2} + 8232 p T^{3} + 253503 p^{2} T^{4} + 8232 p^{4} T^{5} - 4436 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 72 T - 9438 T^{2} - 65088 T^{3} + 131199947 T^{4} - 65088 p^{3} T^{5} - 9438 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 308 T + 50342 T^{2} - 6217904 T^{3} + 679962307 T^{4} - 6217904 p^{3} T^{5} + 50342 p^{6} T^{6} - 308 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 80 T + 18626 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 384 T + 54538 T^{2} + 12801024 T^{3} + 3353389347 T^{4} + 12801024 p^{3} T^{5} + 54538 p^{6} T^{6} + 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 536 T + 128278 T^{2} + 30933632 T^{3} + 8168593627 T^{4} + 30933632 p^{3} T^{5} + 128278 p^{6} T^{6} + 536 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 756 T + 278276 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 400 T + 142566 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 312 T - 124838 T^{2} + 4535232 T^{3} + 28479079683 T^{4} + 4535232 p^{3} T^{5} - 124838 p^{6} T^{6} + 312 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 52 T - 239278 T^{2} + 2900144 T^{3} + 35988363787 T^{4} + 2900144 p^{3} T^{5} - 239278 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 864 T + 2534 p T^{2} + 160904448 T^{3} + 160901924475 T^{4} + 160904448 p^{3} T^{5} + 2534 p^{7} T^{6} + 864 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1416 T + 1066392 T^{2} + 686338032 T^{3} + 374460114599 T^{4} + 686338032 p^{3} T^{5} + 1066392 p^{6} T^{6} + 1416 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 144 T - 529526 T^{2} - 7382016 T^{3} + 206093264907 T^{4} - 7382016 p^{3} T^{5} - 529526 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1524 T + 1208266 T^{2} + 1524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 744 T + 312240 T^{2} - 399333072 T^{3} - 308445380785 T^{4} - 399333072 p^{3} T^{5} + 312240 p^{6} T^{6} + 744 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 976 T + 419842 T^{2} - 442463744 T^{3} - 428939059229 T^{4} - 442463744 p^{3} T^{5} + 419842 p^{6} T^{6} + 976 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 108 T - 1318772 T^{2} - 8586216 T^{3} + 1264855900719 T^{4} - 8586216 p^{3} T^{5} - 1318772 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 744 T + 432480 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−8.125442378427555823913787385517, −7.61271488303290067608858973158, −7.41637051508346668859630111823, −7.39283702176217195031815127889, −7.26670133732136683151881204844, −7.02608986549800624459402142989, −6.29521069487828723073820339072, −6.17124921474099116521944474638, −5.80185397245794999506865940588, −5.77383121493717657293982455716, −5.58928467410931036836272388365, −4.81938900198062111148271417581, −4.75961861815232816516979655597, −4.47965397909225176127969127038, −4.09876864245014565297127878991, −4.01760441731965950351289828439, −3.61289545260754700982011518033, −3.27042330189544858251896286696, −3.04279900668972029271334906733, −2.70990747825762971402160847545, −2.66139985896971616853909364561, −1.72792034029123680582538612145, −1.25720429808005395664875326373, −1.12687118100300262988279904597, −0.16420879536044292084214607426,
0.16420879536044292084214607426, 1.12687118100300262988279904597, 1.25720429808005395664875326373, 1.72792034029123680582538612145, 2.66139985896971616853909364561, 2.70990747825762971402160847545, 3.04279900668972029271334906733, 3.27042330189544858251896286696, 3.61289545260754700982011518033, 4.01760441731965950351289828439, 4.09876864245014565297127878991, 4.47965397909225176127969127038, 4.75961861815232816516979655597, 4.81938900198062111148271417581, 5.58928467410931036836272388365, 5.77383121493717657293982455716, 5.80185397245794999506865940588, 6.17124921474099116521944474638, 6.29521069487828723073820339072, 7.02608986549800624459402142989, 7.26670133732136683151881204844, 7.39283702176217195031815127889, 7.41637051508346668859630111823, 7.61271488303290067608858973158, 8.125442378427555823913787385517
