| L(s) = 1 | − 3·4-s + 4·5-s + 6·9-s + 6·11-s + 4·16-s − 24·19-s − 12·20-s + 5·25-s + 20·29-s − 16·31-s − 18·36-s − 4·41-s − 18·44-s + 24·45-s + 49-s + 24·55-s + 8·59-s − 20·61-s − 9·64-s − 20·71-s + 72·76-s + 14·79-s + 16·80-s + 27·81-s + 72·89-s − 96·95-s + 36·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.78·5-s + 2·9-s + 1.80·11-s + 16-s − 5.50·19-s − 2.68·20-s + 25-s + 3.71·29-s − 2.87·31-s − 3·36-s − 0.624·41-s − 2.71·44-s + 3.57·45-s + 1/7·49-s + 3.23·55-s + 1.04·59-s − 2.56·61-s − 9/8·64-s − 2.37·71-s + 8.25·76-s + 1.57·79-s + 1.78·80-s + 3·81-s + 7.63·89-s − 9.84·95-s + 3.61·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.997681444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997681444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 141 T^{2} + 12992 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \) |
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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
−8.776174752896853348922202379219, −8.357981900533143131951681217761, −7.953314737700527273525967080577, −7.951969760316942005286915588588, −7.44646811626751569166101267025, −6.98279660966498628837162444446, −6.81884260522802219274664819798, −6.59817826399178389796430865510, −6.35333956850444061378629582218, −6.20464139653574160431434222451, −6.04627201130400932013796927524, −5.73753709263598194581812969663, −5.01148511360456842377543932484, −4.94671168245845397176769528962, −4.51766033524863926662016071571, −4.37602846155523541962820697272, −4.35820986211682045510658817944, −3.93387277418203653643475218529, −3.56554701571254983224344504052, −3.24072600716166047601148258376, −2.38324604528608972742369904198, −2.02170218841923569028057418465, −1.87664749984015569127340094998, −1.55482727834239999611678247348, −0.65912522498567746488438829373,
0.65912522498567746488438829373, 1.55482727834239999611678247348, 1.87664749984015569127340094998, 2.02170218841923569028057418465, 2.38324604528608972742369904198, 3.24072600716166047601148258376, 3.56554701571254983224344504052, 3.93387277418203653643475218529, 4.35820986211682045510658817944, 4.37602846155523541962820697272, 4.51766033524863926662016071571, 4.94671168245845397176769528962, 5.01148511360456842377543932484, 5.73753709263598194581812969663, 6.04627201130400932013796927524, 6.20464139653574160431434222451, 6.35333956850444061378629582218, 6.59817826399178389796430865510, 6.81884260522802219274664819798, 6.98279660966498628837162444446, 7.44646811626751569166101267025, 7.951969760316942005286915588588, 7.953314737700527273525967080577, 8.357981900533143131951681217761, 8.776174752896853348922202379219
