| L(s) = 1 | + 2·5-s − 3·9-s + 2·17-s + 11·25-s − 6·37-s + 32·41-s − 6·45-s − 2·49-s + 42·53-s − 42·61-s + 30·73-s + 4·85-s − 26·89-s − 14·101-s − 10·109-s + 3·121-s + 38·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 0.485·17-s + 11/5·25-s − 0.986·37-s + 4.99·41-s − 0.894·45-s − 2/7·49-s + 5.76·53-s − 5.37·61-s + 3.51·73-s + 0.433·85-s − 2.75·89-s − 1.39·101-s − 0.957·109-s + 3/11·121-s + 3.39·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.365465310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.365465310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 21 T^{2} - 88 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 19 T^{2} - 600 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 91 T^{2} + 6072 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 21 T + 200 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.48245719645510557073892918207, −7.37135548281743268643585064414, −7.04232450335560272296397628603, −6.91296687805008168177789066824, −6.67089949470635481937626698001, −6.20917660725994702488636681610, −6.18320374690058594629262487000, −5.80757646793603827737710735017, −5.61520819974360073009925296019, −5.46075418081510686421819452661, −5.42960561111130695834852851574, −4.93306909201831707425399751582, −4.57803508742264549528698043728, −4.33049066356088925436563635132, −4.11882512265963944419009564909, −4.00246640870059927783486114069, −3.24538114315672058162579459365, −3.09771292707196090308183664817, −3.08476327213726297135428719840, −2.41907790935511280199417700708, −2.26302390163103273836502693904, −2.21894742463176567994385859965, −1.21092481573229219587400814192, −1.17360475900705210876994840932, −0.57546424641085421603792033567,
0.57546424641085421603792033567, 1.17360475900705210876994840932, 1.21092481573229219587400814192, 2.21894742463176567994385859965, 2.26302390163103273836502693904, 2.41907790935511280199417700708, 3.08476327213726297135428719840, 3.09771292707196090308183664817, 3.24538114315672058162579459365, 4.00246640870059927783486114069, 4.11882512265963944419009564909, 4.33049066356088925436563635132, 4.57803508742264549528698043728, 4.93306909201831707425399751582, 5.42960561111130695834852851574, 5.46075418081510686421819452661, 5.61520819974360073009925296019, 5.80757646793603827737710735017, 6.18320374690058594629262487000, 6.20917660725994702488636681610, 6.67089949470635481937626698001, 6.91296687805008168177789066824, 7.04232450335560272296397628603, 7.37135548281743268643585064414, 7.48245719645510557073892918207
