| L(s) = 1 | − 4-s − 3·16-s + 16·19-s + 10·25-s + 16·31-s − 28·49-s + 4·61-s + 18·64-s − 16·76-s − 32·79-s − 10·100-s − 56·109-s + 44·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3/4·16-s + 3.67·19-s + 2·25-s + 2.87·31-s − 4·49-s + 0.512·61-s + 9/4·64-s − 1.83·76-s − 3.60·79-s − 100-s − 5.36·109-s + 4·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899298054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899298054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T^{2} + p^{2} T^{4} )^{2}( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} ) \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 14 T^{2} - 93 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2}( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 29 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 14 T^{2} - 2013 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 86 T^{2} + 4587 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{4}( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + 16 T + p T^{2} )^{4}( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2}( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.89179685661454298787639906301, −4.78370008022067975672658132756, −4.76831029399161634698749427356, −4.58893464093038212514923734712, −4.56269239311471389251695290247, −4.42213957632051857657057412597, −4.14324830882665351523652488252, −3.90312531084741822078495608840, −3.67835556500156124020294831701, −3.61156036583542118361405869823, −3.55162444808671451560058563729, −3.34511668680752012295447771563, −3.06722801209360359387416902961, −2.87646697389536358768703109684, −2.80219041568417633658496251795, −2.79190737081343033503970162080, −2.66906928659380283514423495942, −2.24502026509057704541396943174, −2.07080736525656654125451378763, −1.79603871943058299405698620999, −1.38114417524495917557848196593, −1.24954117323933899279344390482, −1.07150790343425971607612439100, −1.00244626266093555539860855246, −0.28641021810243752565999534307,
0.28641021810243752565999534307, 1.00244626266093555539860855246, 1.07150790343425971607612439100, 1.24954117323933899279344390482, 1.38114417524495917557848196593, 1.79603871943058299405698620999, 2.07080736525656654125451378763, 2.24502026509057704541396943174, 2.66906928659380283514423495942, 2.79190737081343033503970162080, 2.80219041568417633658496251795, 2.87646697389536358768703109684, 3.06722801209360359387416902961, 3.34511668680752012295447771563, 3.55162444808671451560058563729, 3.61156036583542118361405869823, 3.67835556500156124020294831701, 3.90312531084741822078495608840, 4.14324830882665351523652488252, 4.42213957632051857657057412597, 4.56269239311471389251695290247, 4.58893464093038212514923734712, 4.76831029399161634698749427356, 4.78370008022067975672658132756, 4.89179685661454298787639906301
Plot not available for L-functions of degree greater than 10.
