| L(s) = 1 | + 1.93e4·9-s − 2.48e4·13-s − 4.95e4·17-s + 1.56e5·25-s + 1.85e5·29-s + 1.10e6·37-s − 7.57e5·41-s + 1.06e7·49-s + 1.32e7·53-s − 1.81e7·61-s − 1.25e7·73-s + 1.95e8·81-s − 1.20e8·89-s + 1.86e8·97-s + 6.24e7·101-s + 4.89e7·109-s − 3.01e8·113-s − 4.81e8·117-s + 4.89e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9.58e8·153-s + 157-s + ⋯ |
| L(s) = 1 | + 2.94·9-s − 0.871·13-s − 0.593·17-s + 2/5·25-s + 0.262·29-s + 0.590·37-s − 0.268·41-s + 1.85·49-s + 1.67·53-s − 1.31·61-s − 0.441·73-s + 4.54·81-s − 1.91·89-s + 2.10·97-s + 0.600·101-s + 0.347·109-s − 1.84·113-s − 2.56·117-s + 2.28·121-s − 1.74·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9852783829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9852783829\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 6448 p T^{2} + 19847614 p^{2} T^{4} - 6448 p^{17} T^{6} + p^{32} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 217696 p^{2} T^{2} + 23540949006 p^{4} T^{4} - 217696 p^{18} T^{6} + p^{32} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 489425524 T^{2} + 121281840487812966 T^{4} - 489425524 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 12440 T + 1670037342 T^{2} + 12440 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 24780 T + 13800826982 T^{2} + 24780 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28089092164 T^{2} + \)\(64\!\cdots\!86\)\( T^{4} - 28089092164 p^{16} T^{6} + p^{32} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 270019273024 T^{2} + \)\(30\!\cdots\!66\)\( T^{4} - 270019273024 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 92724 T + 800033810966 T^{2} - 92724 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 405106859764 T^{2} - \)\(50\!\cdots\!14\)\( T^{4} - 405106859764 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 552920 T + 3669742026942 T^{2} - 552920 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 378744 T + 7437652700126 T^{2} + 378744 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 33641189179504 T^{2} + \)\(51\!\cdots\!06\)\( T^{4} - 33641189179504 p^{16} T^{6} + p^{32} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 35823614614144 T^{2} + \)\(14\!\cdots\!26\)\( T^{4} - 35823614614144 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 6611640 T + 43356528082622 T^{2} - 6611640 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 415010066657284 T^{2} + \)\(84\!\cdots\!46\)\( T^{4} - 415010066657284 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 9084736 T + 310132120029486 T^{2} + 9084736 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 767036284974064 T^{2} + \)\(46\!\cdots\!86\)\( T^{4} - 767036284974064 p^{16} T^{6} + p^{32} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 423108310973044 T^{2} + \)\(24\!\cdots\!26\)\( T^{4} - 423108310973044 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 6275740 T + 1619013167125062 T^{2} + 6275740 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1502171306813756 T^{2} + \)\(51\!\cdots\!26\)\( T^{4} + 1502171306813756 p^{16} T^{6} + p^{32} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2406153547014064 T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - 2406153547014064 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 60010764 T + 7158572584170086 T^{2} + 60010764 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 93177980 T + 8654801913524022 T^{2} - 93177980 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−7.17755813612007026330988926129, −6.94552412849369350158616500321, −6.57418909350737666046664805212, −6.32195285360300970325061959180, −6.04983823049904445264313479866, −5.70586443997342088549716532554, −5.55723188373428145772973415773, −5.07016440783004658124483129849, −4.73564086793620713122985014638, −4.69143429248154534105047072893, −4.37582674203163927645900570177, −4.29262129003873502923667468696, −3.76106374206687187727842546989, −3.75647464039187206851678775484, −3.36463604126253175656364315760, −2.81091684180345436387644332350, −2.70780531382842103434800800182, −2.23548661102951221944033216043, −2.04592475546605173717862233524, −1.78283595886428605845944903854, −1.40964222940817765804664363602, −1.11688141268140568450261901426, −0.794094739044105958177278662128, −0.71783285309818701180842977299, −0.081598118392326516237021539048,
0.081598118392326516237021539048, 0.71783285309818701180842977299, 0.794094739044105958177278662128, 1.11688141268140568450261901426, 1.40964222940817765804664363602, 1.78283595886428605845944903854, 2.04592475546605173717862233524, 2.23548661102951221944033216043, 2.70780531382842103434800800182, 2.81091684180345436387644332350, 3.36463604126253175656364315760, 3.75647464039187206851678775484, 3.76106374206687187727842546989, 4.29262129003873502923667468696, 4.37582674203163927645900570177, 4.69143429248154534105047072893, 4.73564086793620713122985014638, 5.07016440783004658124483129849, 5.55723188373428145772973415773, 5.70586443997342088549716532554, 6.04983823049904445264313479866, 6.32195285360300970325061959180, 6.57418909350737666046664805212, 6.94552412849369350158616500321, 7.17755813612007026330988926129
