| L(s) = 1 | + 4·3-s − 5-s + 6·9-s + 4·13-s − 4·15-s + 25-s − 4·27-s + 8·31-s + 4·37-s + 16·39-s + 12·41-s + 20·43-s − 6·45-s − 10·49-s − 12·53-s − 4·65-s − 4·67-s + 24·71-s + 4·75-s − 16·79-s − 37·81-s − 12·83-s − 12·89-s + 32·93-s + 12·107-s + 16·111-s + 24·117-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.447·5-s + 2·9-s + 1.10·13-s − 1.03·15-s + 1/5·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 2.56·39-s + 1.87·41-s + 3.04·43-s − 0.894·45-s − 1.42·49-s − 1.64·53-s − 0.496·65-s − 0.488·67-s + 2.84·71-s + 0.461·75-s − 1.80·79-s − 4.11·81-s − 1.31·83-s − 1.27·89-s + 3.31·93-s + 1.16·107-s + 1.51·111-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.270950819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.270950819\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.249279365224539237566485048625, −8.740293324354028093320034492248, −8.550149092441660717391493732033, −7.950176007435402201982724906846, −7.71273110823499542846308181544, −7.33279621647751156873215628670, −6.27087624192875571051851265258, −6.14181160684855424693200797613, −5.25974231276140424950498810771, −4.19097847493160731891690631788, −4.12250433368686324236236368171, −3.32887958847832114233485788826, −2.76929890617261215013507568311, −2.41662881718811067600819256762, −1.26575059491370931038306222328,
1.26575059491370931038306222328, 2.41662881718811067600819256762, 2.76929890617261215013507568311, 3.32887958847832114233485788826, 4.12250433368686324236236368171, 4.19097847493160731891690631788, 5.25974231276140424950498810771, 6.14181160684855424693200797613, 6.27087624192875571051851265258, 7.33279621647751156873215628670, 7.71273110823499542846308181544, 7.950176007435402201982724906846, 8.550149092441660717391493732033, 8.740293324354028093320034492248, 9.249279365224539237566485048625
