| L(s) = 1 | − 3·9-s + 4·13-s + 10·25-s + 20·37-s + 2·49-s − 28·61-s + 20·73-s + 9·81-s − 28·97-s + 4·109-s − 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 2·25-s + 3.28·37-s + 2/7·49-s − 3.58·61-s + 2.34·73-s + 81-s − 2.84·97-s + 0.383·109-s − 1.10·117-s − 2·121-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277028929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277028929\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62319681634606304500547076579, −12.48156897690249202858230097469, −11.68598738069570225342057783170, −11.25165812233945499027023204816, −10.84893670739889342900815061984, −10.60278849114699696559566293376, −9.708184890174179423079431268622, −9.161734831488399239895785093245, −8.937434081000418110369817580115, −8.041862585418433570099595947456, −8.031716069625640784140838134814, −7.11832588898137780415786460288, −6.30023882575175586006277492091, −6.18228802374312921582724731063, −5.38158919701726417869764232295, −4.70291808743084588513594320992, −4.02421232543024795597776629105, −3.10170155764302832632860294951, −2.60182485234272276047494052157, −1.13039910110545379294649008287,
1.13039910110545379294649008287, 2.60182485234272276047494052157, 3.10170155764302832632860294951, 4.02421232543024795597776629105, 4.70291808743084588513594320992, 5.38158919701726417869764232295, 6.18228802374312921582724731063, 6.30023882575175586006277492091, 7.11832588898137780415786460288, 8.031716069625640784140838134814, 8.041862585418433570099595947456, 8.937434081000418110369817580115, 9.161734831488399239895785093245, 9.708184890174179423079431268622, 10.60278849114699696559566293376, 10.84893670739889342900815061984, 11.25165812233945499027023204816, 11.68598738069570225342057783170, 12.48156897690249202858230097469, 12.62319681634606304500547076579
