| L(s) = 1 | + 7-s + 3·13-s − 3·19-s + 2·25-s + 3·31-s − 37-s − 43-s + 67-s + 3·73-s + 79-s + 3·91-s − 109-s − 2·121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + 179-s + ⋯ |
| L(s) = 1 | + 7-s + 3·13-s − 3·19-s + 2·25-s + 3·31-s − 37-s − 43-s + 67-s + 3·73-s + 79-s + 3·91-s − 109-s − 2·121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s + 2·175-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.749877911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749877911\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.979441483544332929766902880955, −8.958941341863152503021898675991, −8.473137596244116041885483283870, −8.442678181910493699750935129733, −8.024350534442175993078598384048, −7.75667065823235830401726484025, −6.67215960365943871774579036999, −6.63852731943742718201619396407, −6.31147507001122015262764068857, −6.26784849691681676767382426068, −5.20732753230937849669996872415, −5.14550878103349304957407066952, −4.57492324675834121825223765147, −4.13632102021669576111453616616, −3.75313942168689720041166296671, −3.36504896752369554051610435511, −2.52660248627938250576905248945, −2.24045872319457536057613420701, −1.26990601036553926798128041157, −1.18152064925248947837004719798,
1.18152064925248947837004719798, 1.26990601036553926798128041157, 2.24045872319457536057613420701, 2.52660248627938250576905248945, 3.36504896752369554051610435511, 3.75313942168689720041166296671, 4.13632102021669576111453616616, 4.57492324675834121825223765147, 5.14550878103349304957407066952, 5.20732753230937849669996872415, 6.26784849691681676767382426068, 6.31147507001122015262764068857, 6.63852731943742718201619396407, 6.67215960365943871774579036999, 7.75667065823235830401726484025, 8.024350534442175993078598384048, 8.442678181910493699750935129733, 8.473137596244116041885483283870, 8.958941341863152503021898675991, 8.979441483544332929766902880955
