| L(s) = 1 | + 2-s − 8-s + 2·11-s − 16-s + 2·17-s − 19-s + 2·22-s − 25-s + 2·34-s − 38-s − 41-s − 2·43-s + 2·49-s − 50-s − 59-s + 64-s + 67-s + 73-s − 82-s + 2·83-s − 2·86-s − 2·88-s + 2·89-s + 97-s + 2·98-s − 4·107-s + 2·113-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s + 2·11-s − 16-s + 2·17-s − 19-s + 2·22-s − 25-s + 2·34-s − 38-s − 41-s − 2·43-s + 2·49-s − 50-s − 59-s + 64-s + 67-s + 73-s − 82-s + 2·83-s − 2·86-s − 2·88-s + 2·89-s + 97-s + 2·98-s − 4·107-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.751280744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751280744\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.909641356401712306348470799423, −9.592029729301271249953825001604, −9.103891131186112088069976664638, −8.965372255574945273045409780896, −8.310543646827297557422856591370, −8.053606816114263555851504370460, −7.55921138106756165977651633892, −6.90020675466280367805861618174, −6.62532180372079981880622406667, −6.15383790416726603854635110445, −5.95224180534742166519737696206, −5.20185141778349170490553110825, −5.09273343030216437140872411861, −4.40307498989974494847252125232, −3.85134957783059151022850704898, −3.62509479241877635208500933010, −3.34667348171865755274524163341, −2.45711534118218481231559164559, −1.80031926845812421099671004847, −1.05614377543293206627987499296,
1.05614377543293206627987499296, 1.80031926845812421099671004847, 2.45711534118218481231559164559, 3.34667348171865755274524163341, 3.62509479241877635208500933010, 3.85134957783059151022850704898, 4.40307498989974494847252125232, 5.09273343030216437140872411861, 5.20185141778349170490553110825, 5.95224180534742166519737696206, 6.15383790416726603854635110445, 6.62532180372079981880622406667, 6.90020675466280367805861618174, 7.55921138106756165977651633892, 8.053606816114263555851504370460, 8.310543646827297557422856591370, 8.965372255574945273045409780896, 9.103891131186112088069976664638, 9.592029729301271249953825001604, 9.909641356401712306348470799423
