| L(s) = 1 | + 4·2-s + 4·4-s + 7·5-s − 16·8-s + 28·10-s + 25·11-s + 98·13-s − 64·16-s + 98·17-s − 119·19-s + 28·20-s + 100·22-s − 122·23-s + 214·25-s + 392·26-s + 146·29-s + 98·31-s − 64·32-s + 392·34-s − 289·37-s − 476·38-s − 112·40-s − 672·41-s + 614·43-s + 100·44-s − 488·46-s + 672·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.626·5-s − 0.707·8-s + 0.885·10-s + 0.685·11-s + 2.09·13-s − 16-s + 1.39·17-s − 1.43·19-s + 0.313·20-s + 0.969·22-s − 1.10·23-s + 1.71·25-s + 2.95·26-s + 0.934·29-s + 0.567·31-s − 0.353·32-s + 1.97·34-s − 1.28·37-s − 2.03·38-s − 0.442·40-s − 2.55·41-s + 2.17·43-s + 0.342·44-s − 1.56·46-s + 2.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(29.34923768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.34923768\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 7 T - 33 p T^{2} + 252 T^{3} + 24046 T^{4} + 252 p^{3} T^{5} - 33 p^{7} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 25 T + 171 T^{2} + 55200 T^{3} - 2397320 T^{4} + 55200 p^{3} T^{5} + 171 p^{6} T^{6} - 25 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 49 T + 3788 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 98 T + 2202 T^{2} + 237552 T^{3} - 16532417 T^{4} + 237552 p^{3} T^{5} + 2202 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 119 T - 2663 T^{2} + 369614 T^{3} + 138870796 T^{4} + 369614 p^{3} T^{5} - 2663 p^{6} T^{6} + 119 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 122 T - 3714 T^{2} - 699792 T^{3} + 16756087 T^{4} - 699792 p^{3} T^{5} - 3714 p^{6} T^{6} + 122 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 73 T + 47746 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 98 T - 14551 T^{2} + 3471846 T^{3} - 590152420 T^{4} + 3471846 p^{3} T^{5} - 14551 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 289 T - 17387 T^{2} - 115022 T^{3} + 3386108842 T^{4} - 115022 p^{3} T^{5} - 17387 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 672 T + 158834 T^{2} - 57189888 T^{3} + 28038541539 T^{4} - 57189888 p^{3} T^{5} + 158834 p^{6} T^{6} - 672 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 375 T - 781 T^{2} + 58630500 T^{3} - 23104532418 T^{4} + 58630500 p^{3} T^{5} - 781 p^{6} T^{6} - 375 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 763 T + 205407 T^{2} + 25938948 T^{3} - 19796216048 T^{4} + 25938948 p^{3} T^{5} + 205407 p^{6} T^{6} - 763 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 406 T - 274558 T^{2} - 5914608 T^{3} + 104132072759 T^{4} - 5914608 p^{3} T^{5} - 274558 p^{6} T^{6} + 406 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1041 T + 270341 T^{2} - 220498374 T^{3} + 245132424828 T^{4} - 220498374 p^{3} T^{5} + 270341 p^{6} T^{6} - 1041 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1652 T + 1360270 T^{2} - 1652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 189 T - 297259 T^{2} - 84115206 T^{3} - 54354622146 T^{4} - 84115206 p^{3} T^{5} - 297259 p^{6} T^{6} + 189 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 524 T - 316753 T^{2} - 206848476 T^{3} - 28794145744 T^{4} - 206848476 p^{3} T^{5} - 316753 p^{6} T^{6} + 524 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 287 T + 781978 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2394 T + 3029186 T^{2} - 3093316128 T^{3} + 2763749199855 T^{4} - 3093316128 p^{3} T^{5} + 3029186 p^{6} T^{6} - 2394 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 63 T + 667132 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.78778476005865526169660235814, −6.36843824990308214691851340835, −6.35364245824370778759414409201, −6.12480816069798079759084467502, −5.97516994416949337470229376732, −5.74517500537795526410812287736, −5.41773293776203551519467865543, −5.15446056322229204974905305603, −5.12137685166680639139669685420, −4.77515735588400412536330839092, −4.32242155744782223640721485438, −4.31505682379000445892581748990, −4.08034593064765230631048702574, −3.48674480269112863648041496033, −3.45752430245226131212810379773, −3.41800086905325087755954998731, −3.35284074710750304906951083001, −2.39431543388357397033533190537, −2.33830701193487504980217854795, −2.23753613263339476819979896610, −1.80823006419047680884452663954, −1.28084542155070143601927160119, −0.910833271251803366621990951643, −0.67242703785661933829848066769, −0.59071862057341245600669537812,
0.59071862057341245600669537812, 0.67242703785661933829848066769, 0.910833271251803366621990951643, 1.28084542155070143601927160119, 1.80823006419047680884452663954, 2.23753613263339476819979896610, 2.33830701193487504980217854795, 2.39431543388357397033533190537, 3.35284074710750304906951083001, 3.41800086905325087755954998731, 3.45752430245226131212810379773, 3.48674480269112863648041496033, 4.08034593064765230631048702574, 4.31505682379000445892581748990, 4.32242155744782223640721485438, 4.77515735588400412536330839092, 5.12137685166680639139669685420, 5.15446056322229204974905305603, 5.41773293776203551519467865543, 5.74517500537795526410812287736, 5.97516994416949337470229376732, 6.12480816069798079759084467502, 6.35364245824370778759414409201, 6.36843824990308214691851340835, 6.78778476005865526169660235814
