| L(s) = 1 | − 100·5-s − 2.78e3·13-s − 380·17-s + 1.25e3·25-s − 2.06e4·37-s + 3.68e4·41-s + 4.53e4·53-s + 7.47e4·61-s + 2.78e5·65-s − 9.62e4·73-s + 1.08e5·81-s + 3.80e4·85-s − 1.97e5·97-s − 4.01e5·101-s − 7.27e5·113-s + 2.93e5·121-s + 4.37e5·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.86e6·169-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 4.56·13-s − 0.318·17-s + 2/5·25-s − 2.47·37-s + 3.42·41-s + 2.21·53-s + 2.57·61-s + 8.16·65-s − 2.11·73-s + 1.84·81-s + 0.570·85-s − 2.12·97-s − 3.91·101-s − 5.35·113-s + 1.82·121-s + 2.50·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 10.4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5527869628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5527869628\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 2 p^{2} T + p^{5} T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - 12098 p^{2} T^{4} + p^{20} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 560853922 T^{4} + p^{20} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 146602 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 1390 T + 966050 T^{2} + 1390 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 190 T + 18050 T^{2} + 190 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4250198 T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 82817669402722 T^{4} + p^{20} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 32750922 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 35581198 T^{2} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 10310 T + 53148050 T^{2} + 10310 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9218 T + p^{5} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 40420440476627918 T^{4} + p^{20} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 85407260265966082 T^{4} + p^{20} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 22690 T + 257418050 T^{2} - 22690 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1429146598 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 18678 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 746452483343412398 T^{4} + p^{20} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 217623202 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 48110 T + 1157286050 T^{2} + 48110 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2651775202 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 2557737354504584722 T^{4} + p^{20} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 5648223282 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 98510 T + 4852110050 T^{2} + 98510 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−9.639749104703631221728279811795, −9.302502587072473171265106723440, −9.257090988970116451057025355208, −8.626417923526845962841878586071, −8.364739801927713419480785493789, −7.86945230861765453108992579788, −7.84940357638755562001592007354, −7.47303353495610373863886479269, −7.01279501566513600636948915139, −6.99572802079066391152478227434, −6.99491443650863416297442032977, −6.12312472288650551408264291420, −5.44974217883828300017346773975, −5.40260139447437979538996356959, −5.06849877393356948645880415044, −4.55263719669582011396006231868, −4.25112417376278024614329650764, −3.94845914941493872525268872500, −3.62918116571160645619895480669, −2.66337866477535571202758743285, −2.54963076510183620081683577310, −2.39083404713299267969880621805, −1.47294453288905145997585829670, −0.42258539250052499854667536783, −0.31255548954118273018838958049,
0.31255548954118273018838958049, 0.42258539250052499854667536783, 1.47294453288905145997585829670, 2.39083404713299267969880621805, 2.54963076510183620081683577310, 2.66337866477535571202758743285, 3.62918116571160645619895480669, 3.94845914941493872525268872500, 4.25112417376278024614329650764, 4.55263719669582011396006231868, 5.06849877393356948645880415044, 5.40260139447437979538996356959, 5.44974217883828300017346773975, 6.12312472288650551408264291420, 6.99491443650863416297442032977, 6.99572802079066391152478227434, 7.01279501566513600636948915139, 7.47303353495610373863886479269, 7.84940357638755562001592007354, 7.86945230861765453108992579788, 8.364739801927713419480785493789, 8.626417923526845962841878586071, 9.257090988970116451057025355208, 9.302502587072473171265106723440, 9.639749104703631221728279811795
