| L(s) = 1 | + 6·3-s + 6·4-s + 4·7-s + 16·9-s + 36·12-s + 12·13-s + 13·16-s + 16·19-s + 24·21-s + 24·27-s + 24·28-s + 8·31-s + 96·36-s − 32·37-s + 72·39-s + 12·43-s + 78·48-s + 8·49-s + 72·52-s + 96·57-s + 28·61-s + 64·63-s + 6·64-s + 32·67-s − 28·73-s + 96·76-s − 16·79-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 3·4-s + 1.51·7-s + 16/3·9-s + 10.3·12-s + 3.32·13-s + 13/4·16-s + 3.67·19-s + 5.23·21-s + 4.61·27-s + 4.53·28-s + 1.43·31-s + 16·36-s − 5.26·37-s + 11.5·39-s + 1.82·43-s + 11.2·48-s + 8/7·49-s + 9.98·52-s + 12.7·57-s + 3.58·61-s + 8.06·63-s + 3/4·64-s + 3.90·67-s − 3.27·73-s + 11.0·76-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(203.8721766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(203.8721766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 2 p T + 20 T^{2} - 16 p T^{3} + 91 T^{4} - 16 p^{2} T^{5} + 20 p^{2} T^{6} - 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 13 | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - 3 T^{2} + 7 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 2 T + 2 T^{2} + 24 T^{3} - 73 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 24 T^{2} + 338 T^{4} + 3504 T^{6} + 29907 T^{8} + 3504 p^{2} T^{10} + 338 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 38 T^{2} + 613 T^{4} + 9614 T^{6} + 189724 T^{8} + 9614 p^{2} T^{10} + 613 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 8 T + 20 T^{2} + 60 T^{3} - 649 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 34 T^{2} - 707 T^{4} + 6154 T^{6} + 1791292 T^{8} + 6154 p^{2} T^{10} - 707 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 4 T + 8 T^{2} - 36 T^{3} - 322 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 16 T + 113 T^{2} + 12 p T^{3} + 44 p T^{4} + 12 p^{2} T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 54 T^{2} + 221 T^{4} + 40554 T^{6} - 627828 T^{8} + 40554 p^{2} T^{10} + 221 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 6 T - 32 T^{2} + 108 T^{3} + 1227 T^{4} + 108 p T^{5} - 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 190 T^{2} + 14535 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 24 T^{2} - 3202 T^{4} - 81456 T^{6} + 70227 T^{8} - 81456 p^{2} T^{10} - 3202 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 16 T + 164 T^{2} - 1308 T^{3} + 10007 T^{4} - 1308 p T^{5} + 164 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 24 T^{2} - 5554 T^{4} + 137904 T^{6} + 8572707 T^{8} + 137904 p^{2} T^{10} - 5554 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 24 T^{2} + 9026 T^{4} - 212016 T^{6} + 16818147 T^{8} - 212016 p^{2} T^{10} + 9026 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 26 T + 2 p T^{2} - 1332 T^{3} - 32593 T^{4} - 1332 p T^{5} + 2 p^{3} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.15723012120239099820501738195, −4.07019092692504431849855798829, −3.79065439613956031934105739799, −3.78407578428341757218907572336, −3.78260630840015414243835592584, −3.50315617218558408560568324167, −3.27496464565921194916625474788, −3.25511159989833851489678258938, −3.21254773800823472425922386035, −3.06197920462959383563449241460, −2.84818396056535751179852148587, −2.77719154191031696864525409098, −2.71970020459623310964014696880, −2.68635752582699905088676815139, −2.16691406197204035765530277100, −2.15757833572314223239431209221, −1.93187962008181106167875503873, −1.93166723806048051554732776823, −1.91753287068196516598806446686, −1.61752166045645740330458852745, −1.49660790495407139632745725750, −1.11630531034486236733417747307, −1.07771421765808142844301377079, −0.950097318339013061968780658579, −0.60135977889579405470420664668,
0.60135977889579405470420664668, 0.950097318339013061968780658579, 1.07771421765808142844301377079, 1.11630531034486236733417747307, 1.49660790495407139632745725750, 1.61752166045645740330458852745, 1.91753287068196516598806446686, 1.93166723806048051554732776823, 1.93187962008181106167875503873, 2.15757833572314223239431209221, 2.16691406197204035765530277100, 2.68635752582699905088676815139, 2.71970020459623310964014696880, 2.77719154191031696864525409098, 2.84818396056535751179852148587, 3.06197920462959383563449241460, 3.21254773800823472425922386035, 3.25511159989833851489678258938, 3.27496464565921194916625474788, 3.50315617218558408560568324167, 3.78260630840015414243835592584, 3.78407578428341757218907572336, 3.79065439613956031934105739799, 4.07019092692504431849855798829, 4.15723012120239099820501738195
Plot not available for L-functions of degree greater than 10.
