| L(s) = 1 | − 3-s − 7-s − 19-s + 21-s + 25-s + 27-s − 31-s + 37-s + 57-s + 3·67-s − 3·73-s − 75-s + 3·79-s − 81-s + 93-s − 103-s − 109-s − 111-s + 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3-s − 7-s − 19-s + 21-s + 25-s + 27-s − 31-s + 37-s + 57-s + 3·67-s − 3·73-s − 75-s + 3·79-s − 81-s + 93-s − 103-s − 109-s − 111-s + 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5064922449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5064922449\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.939477619713863124558593496778, −9.748662099343130895009444063186, −9.130614759514954504775436160644, −8.980126425862972819521013077479, −8.281749378928719725311835834235, −8.186573575455531062631387939332, −7.37261993305292518376043035885, −7.11567463233160990658616053117, −6.52687517333919193066083666250, −6.34817592919373974164104366699, −6.00903246310989195630961886384, −5.40763790714627863062203672965, −5.06693735760378747833297565057, −4.62810385845283785296937393381, −3.92432607490802896212479042824, −3.64638772608634540653026694452, −2.84208619443475533796285493813, −2.53856528707490066149547813366, −1.64818073051332115170411506890, −0.64355509445106793092483028764,
0.64355509445106793092483028764, 1.64818073051332115170411506890, 2.53856528707490066149547813366, 2.84208619443475533796285493813, 3.64638772608634540653026694452, 3.92432607490802896212479042824, 4.62810385845283785296937393381, 5.06693735760378747833297565057, 5.40763790714627863062203672965, 6.00903246310989195630961886384, 6.34817592919373974164104366699, 6.52687517333919193066083666250, 7.11567463233160990658616053117, 7.37261993305292518376043035885, 8.186573575455531062631387939332, 8.281749378928719725311835834235, 8.980126425862972819521013077479, 9.130614759514954504775436160644, 9.748662099343130895009444063186, 9.939477619713863124558593496778
