| L(s) = 1 | − 2·4-s + 9-s + 16-s − 2·36-s − 49-s + 2·64-s + 6·79-s + 6·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 9-s + 16-s − 2·36-s − 49-s + 2·64-s + 6·79-s + 6·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 2·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8582304803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8582304803\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.37301883309346037794358303309, −6.30658264706871980479715539672, −5.95283237038271013371555721113, −5.75477604392080847559428312059, −5.72529640518129001104715595277, −5.29992361955301165075739176543, −4.96105797523618330106837590952, −4.94144368048906090134216616970, −4.85326081106356257485626276436, −4.52311464367830147949812882282, −4.42900797732949013418075560640, −4.39371332324238231900013737765, −3.81682438157950506737165680060, −3.75135460286382860063789212531, −3.55876433443509702611320613231, −3.42105061971231433185592816558, −3.08924180433997133229528868416, −2.87266190732957214268904877643, −2.34687982088113549759934473220, −2.02348480493647955834351647462, −1.92718824442053268999928333034, −1.92296107866469568941651753027, −1.00047695815364692365318543008, −0.961235174389485988199827798295, −0.54878027015358715741982972381,
0.54878027015358715741982972381, 0.961235174389485988199827798295, 1.00047695815364692365318543008, 1.92296107866469568941651753027, 1.92718824442053268999928333034, 2.02348480493647955834351647462, 2.34687982088113549759934473220, 2.87266190732957214268904877643, 3.08924180433997133229528868416, 3.42105061971231433185592816558, 3.55876433443509702611320613231, 3.75135460286382860063789212531, 3.81682438157950506737165680060, 4.39371332324238231900013737765, 4.42900797732949013418075560640, 4.52311464367830147949812882282, 4.85326081106356257485626276436, 4.94144368048906090134216616970, 4.96105797523618330106837590952, 5.29992361955301165075739176543, 5.72529640518129001104715595277, 5.75477604392080847559428312059, 5.95283237038271013371555721113, 6.30658264706871980479715539672, 6.37301883309346037794358303309
