| L(s) = 1 | − 4·3-s + 6·9-s + 16·13-s + 2·25-s + 4·27-s − 16·37-s − 64·39-s − 4·49-s − 8·75-s − 37·81-s + 24·83-s + 24·107-s + 64·111-s + 96·117-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 4.43·13-s + 2/5·25-s + 0.769·27-s − 2.63·37-s − 10.2·39-s − 4/7·49-s − 0.923·75-s − 4.11·81-s + 2.63·83-s + 2.32·107-s + 6.07·111-s + 8.87·117-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196181844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196181844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.54121728581489532861533817085, −6.10546299602104740359722387681, −6.10088357053194398129776068516, −6.05507641527199802347694699498, −5.80564522148800512674059883441, −5.56260681412147814521675418759, −5.26122637874589779351830328624, −5.11420301377822517693419665067, −5.09859062773033888009641792750, −4.55366764392720175156261290797, −4.53361677181680364980005836111, −4.23281155780174363063426653620, −3.89809857066315880337094612699, −3.61135480089446761797195890840, −3.56546693166687409712975612538, −3.26580603282515368443517706283, −3.02518549762816683954959749108, −2.92644115698893507746360006882, −2.05469728873766058436072817781, −1.98323307759391271691971399777, −1.76928511473997099880095806811, −1.18568880018780409129030275935, −0.975531391699106144688981089592, −0.900650573084941328723160121838, −0.28679037645353120306288843094,
0.28679037645353120306288843094, 0.900650573084941328723160121838, 0.975531391699106144688981089592, 1.18568880018780409129030275935, 1.76928511473997099880095806811, 1.98323307759391271691971399777, 2.05469728873766058436072817781, 2.92644115698893507746360006882, 3.02518549762816683954959749108, 3.26580603282515368443517706283, 3.56546693166687409712975612538, 3.61135480089446761797195890840, 3.89809857066315880337094612699, 4.23281155780174363063426653620, 4.53361677181680364980005836111, 4.55366764392720175156261290797, 5.09859062773033888009641792750, 5.11420301377822517693419665067, 5.26122637874589779351830328624, 5.56260681412147814521675418759, 5.80564522148800512674059883441, 6.05507641527199802347694699498, 6.10088357053194398129776068516, 6.10546299602104740359722387681, 6.54121728581489532861533817085
