L(s) = 1 | + 4·3-s + 6·9-s + 12·17-s − 8·19-s − 4·27-s + 12·41-s + 20·43-s − 10·49-s + 48·51-s − 32·57-s + 24·59-s − 4·67-s − 4·73-s − 37·81-s − 12·83-s − 12·89-s − 4·97-s + 12·107-s + 12·113-s − 22·121-s + 48·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s − 40·147-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·9-s + 2.91·17-s − 1.83·19-s − 0.769·27-s + 1.87·41-s + 3.04·43-s − 1.42·49-s + 6.72·51-s − 4.23·57-s + 3.12·59-s − 0.488·67-s − 0.468·73-s − 4.11·81-s − 1.31·83-s − 1.27·89-s − 0.406·97-s + 1.16·107-s + 1.12·113-s − 2·121-s + 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.29·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.633451596\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.633451596\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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, −8.550149092441660717391493732033, −8.414042470938792927705837581638, −7.950176007435402201982724906846, −7.38409000222510611898226618037, −7.33279621647751156873215628670, −6.14181160684855424693200797613, −5.85740317846722582719715899176, −5.25974231276140424950498810771, −4.19097847493160731891690631788, −3.90074996505307731038192172560, −3.32887958847832114233485788826, −2.58321256178540655780606724063, −2.41662881718811067600819256762, −1.26575059491370931038306222328,
1.26575059491370931038306222328, 2.41662881718811067600819256762, 2.58321256178540655780606724063, 3.32887958847832114233485788826, 3.90074996505307731038192172560, 4.19097847493160731891690631788, 5.25974231276140424950498810771, 5.85740317846722582719715899176, 6.14181160684855424693200797613, 7.33279621647751156873215628670, 7.38409000222510611898226618037, 7.950176007435402201982724906846, 8.414042470938792927705837581638, 8.550149092441660717391493732033, 9.249279365224539237566485048625