| L(s) = 1 | + 20·7-s − 40·13-s + 23·16-s + 32·31-s − 40·37-s − 40·43-s + 200·49-s − 232·61-s − 280·67-s − 220·73-s − 800·91-s + 20·97-s + 140·103-s + 460·112-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 800·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 20/7·7-s − 3.07·13-s + 1.43·16-s + 1.03·31-s − 1.08·37-s − 0.930·43-s + 4.08·49-s − 3.80·61-s − 4.17·67-s − 3.01·73-s − 8.79·91-s + 0.206·97-s + 1.35·103-s + 4.10·112-s + 0.132·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.73·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.325647209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325647209\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 23 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 20 T + 200 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 144322 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 517762 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 568 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 20 T + 200 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2362 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 20 T + 200 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 8681918 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3037282 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4712 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 140 T + 9800 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 6082 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 110 T + 6050 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 30948638 T^{4} + p^{8} T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.642219398684155678021298109042, −8.456258380212462807901334575601, −7.988489409076462213368219161841, −7.82121684987129264121197613712, −7.67283350517422614909977684391, −7.54342383388645393809122842642, −7.33790850534684631950847113920, −6.83973901348294110471160470731, −6.77837855407766225570091007752, −5.99563861749301135991868899912, −5.81881629518232746121357262978, −5.79502378309746783549494268302, −5.10563886259127585654566772553, −4.88776172701132919906334581612, −4.87623887485188403368049386805, −4.59889006024869311985489580048, −4.27673585871707364167827270216, −3.93210166117685329323144448654, −3.09399946290191885360700143329, −2.83506078404777647726742120102, −2.81469025955103456622137968752, −1.79091988433928630028158836725, −1.72492864183256720987756864315, −1.47085158175037687755412747129, −0.40182907754349849224114239295,
0.40182907754349849224114239295, 1.47085158175037687755412747129, 1.72492864183256720987756864315, 1.79091988433928630028158836725, 2.81469025955103456622137968752, 2.83506078404777647726742120102, 3.09399946290191885360700143329, 3.93210166117685329323144448654, 4.27673585871707364167827270216, 4.59889006024869311985489580048, 4.87623887485188403368049386805, 4.88776172701132919906334581612, 5.10563886259127585654566772553, 5.79502378309746783549494268302, 5.81881629518232746121357262978, 5.99563861749301135991868899912, 6.77837855407766225570091007752, 6.83973901348294110471160470731, 7.33790850534684631950847113920, 7.54342383388645393809122842642, 7.67283350517422614909977684391, 7.82121684987129264121197613712, 7.988489409076462213368219161841, 8.456258380212462807901334575601, 8.642219398684155678021298109042
