| L(s) = 1 | − 2·3-s − 8·7-s + 9-s − 4·13-s − 4·19-s + 16·21-s − 6·25-s + 4·27-s − 20·37-s + 8·39-s − 12·43-s + 34·49-s + 8·57-s − 4·61-s − 8·63-s − 20·67-s + 28·73-s + 12·75-s − 16·79-s − 11·81-s + 32·91-s − 4·97-s − 8·103-s + 12·109-s + 40·111-s − 4·117-s − 18·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3.02·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s + 3.49·21-s − 6/5·25-s + 0.769·27-s − 3.28·37-s + 1.28·39-s − 1.82·43-s + 34/7·49-s + 1.05·57-s − 0.512·61-s − 1.00·63-s − 2.44·67-s + 3.27·73-s + 1.38·75-s − 1.80·79-s − 1.22·81-s + 3.35·91-s − 0.406·97-s − 0.788·103-s + 1.14·109-s + 3.79·111-s − 0.369·117-s − 1.63·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.759776534720553396196404795081, −8.635939205168057076823393046651, −7.62957697295837367057177448880, −6.97464122003923989664965494039, −6.80509880453900742773724680138, −6.33050260025483473718379487161, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −4.84373431553687523132526670248, −3.98566652809305192679300882554, −3.36730269661969347724618284285, −2.97266024838274409285785943283, −1.98277685475006180754048515240, 0, 0,
1.98277685475006180754048515240, 2.97266024838274409285785943283, 3.36730269661969347724618284285, 3.98566652809305192679300882554, 4.84373431553687523132526670248, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.33050260025483473718379487161, 6.80509880453900742773724680138, 6.97464122003923989664965494039, 7.62957697295837367057177448880, 8.635939205168057076823393046651, 8.759776534720553396196404795081
