L(s) = 1 | + 2·5-s − 2·13-s − 4·17-s + 2·25-s + 6·29-s + 10·37-s + 14·49-s + 18·53-s + 2·61-s − 4·65-s − 8·85-s − 16·97-s + 18·101-s − 14·109-s − 32·113-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.554·13-s − 0.970·17-s + 2/5·25-s + 1.11·29-s + 1.64·37-s + 2·49-s + 2.47·53-s + 0.256·61-s − 0.496·65-s − 0.867·85-s − 1.62·97-s + 1.79·101-s − 1.34·109-s − 3.01·113-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.337846143\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.337846143\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.398604899006587194836044086120, −8.290747480528181667134476737145, −7.77021233353889683937758206598, −7.37092470628641945433579464327, −6.85082649107705031559341590357, −6.83822911964323793443502931778, −6.37428745347641656753532018955, −5.81564143714477312510900830336, −5.62593666383534706084644288491, −5.34800209439039141221458181245, −4.66641835114226056963150161311, −4.49292597215270318372492887236, −4.03927060116575121654180568584, −3.66125950605366325040991454388, −2.83777641314297855916474071490, −2.61887592908180619358842776956, −2.30689092388352264571914673880, −1.76407406020900387296644921324, −1.04451152830566912102537416791, −0.56234470100350292272834467666,
0.56234470100350292272834467666, 1.04451152830566912102537416791, 1.76407406020900387296644921324, 2.30689092388352264571914673880, 2.61887592908180619358842776956, 2.83777641314297855916474071490, 3.66125950605366325040991454388, 4.03927060116575121654180568584, 4.49292597215270318372492887236, 4.66641835114226056963150161311, 5.34800209439039141221458181245, 5.62593666383534706084644288491, 5.81564143714477312510900830336, 6.37428745347641656753532018955, 6.83822911964323793443502931778, 6.85082649107705031559341590357, 7.37092470628641945433579464327, 7.77021233353889683937758206598, 8.290747480528181667134476737145, 8.398604899006587194836044086120