| L(s) = 1 | − 4-s − 2·7-s − 2·9-s − 2·13-s + 16-s + 2·19-s + 4·25-s + 2·28-s + 2·31-s + 2·36-s − 2·37-s + 3·49-s + 2·52-s − 2·61-s + 4·63-s − 2·64-s − 4·67-s + 2·73-s − 2·76-s − 2·79-s + 3·81-s + 4·91-s − 2·97-s − 4·100-s + 2·103-s − 2·112-s + 4·117-s + ⋯ |
| L(s) = 1 | − 4-s − 2·7-s − 2·9-s − 2·13-s + 16-s + 2·19-s + 4·25-s + 2·28-s + 2·31-s + 2·36-s − 2·37-s + 3·49-s + 2·52-s − 2·61-s + 4·63-s − 2·64-s − 4·67-s + 2·73-s − 2·76-s − 2·79-s + 3·81-s + 4·91-s − 2·97-s − 4·100-s + 2·103-s − 2·112-s + 4·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1450952368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1450952368\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.216889091870554163749756343538, −9.105257492640903248615743272686, −9.028142894516585149340121942550, −8.656001310041740246012097114734, −8.419824572179928034669012973985, −8.244543117062685754216292349168, −7.63639696742692720184301075209, −7.50380806710069990355949851386, −7.18369849732048065400585758462, −7.00874152878573842366442728339, −6.56762273525719702520866453237, −6.44965633872900794959990974483, −5.86237701988828560970330625284, −5.86192943407996183961397074661, −5.35801835167399409429081676615, −5.16449676002361227003296168042, −4.86565227648055727757135798497, −4.43400830213849649691932529700, −4.43329433667706348483214198311, −3.33240206634039053429583645498, −3.25837212886460724700325050683, −3.06862958808679531741982289930, −2.86926749676441456327653966422, −2.45801410805512419283023089293, −1.15003496418143021220767078789,
1.15003496418143021220767078789, 2.45801410805512419283023089293, 2.86926749676441456327653966422, 3.06862958808679531741982289930, 3.25837212886460724700325050683, 3.33240206634039053429583645498, 4.43329433667706348483214198311, 4.43400830213849649691932529700, 4.86565227648055727757135798497, 5.16449676002361227003296168042, 5.35801835167399409429081676615, 5.86192943407996183961397074661, 5.86237701988828560970330625284, 6.44965633872900794959990974483, 6.56762273525719702520866453237, 7.00874152878573842366442728339, 7.18369849732048065400585758462, 7.50380806710069990355949851386, 7.63639696742692720184301075209, 8.244543117062685754216292349168, 8.419824572179928034669012973985, 8.656001310041740246012097114734, 9.028142894516585149340121942550, 9.105257492640903248615743272686, 9.216889091870554163749756343538
