| L(s) = 1 | + 4·3-s + 8·9-s + 16·11-s + 12·13-s + 12·27-s + 64·33-s + 20·37-s + 48·39-s − 48·47-s − 24·49-s − 8·59-s + 36·61-s + 23·81-s − 32·97-s + 128·99-s − 8·107-s − 12·109-s + 80·111-s + 96·117-s + 128·121-s + 127-s + 131-s + 137-s + 139-s − 192·141-s + 192·143-s − 96·147-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 4.82·11-s + 3.32·13-s + 2.30·27-s + 11.1·33-s + 3.28·37-s + 7.68·39-s − 7.00·47-s − 3.42·49-s − 1.04·59-s + 4.60·61-s + 23/9·81-s − 3.24·97-s + 12.8·99-s − 0.773·107-s − 1.14·109-s + 7.59·111-s + 8.87·117-s + 11.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 16.1·141-s + 16.0·143-s − 7.91·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.24901668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.24901668\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 7822 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−7.68933248957775631775006747189, −6.79976355877924370065118598333, −6.79204294650657684779798179717, −6.74991374734458807347112387877, −6.61497479796695823445518913407, −6.54478385612963626388613185951, −6.07699692492010018418819055165, −5.99385731499379511746783184476, −5.81432259883392504326117803054, −5.14063041584431801867210174583, −4.86010269480415349883814016780, −4.68535793496915786939857257863, −4.23980841406719920934813001860, −3.98325317501609755883990210461, −3.91242906039749612943603470561, −3.68409460473347040019768056594, −3.52454458944618173453709643275, −3.21700579475301496037635134371, −3.01466498705365665386079244742, −2.72249543771234512317126561289, −1.98712168246608165656032518951, −1.67348791963502202712110248894, −1.52773553456272051635098804321, −1.19098019162117405364092969146, −1.07953960950380207640986070222,
1.07953960950380207640986070222, 1.19098019162117405364092969146, 1.52773553456272051635098804321, 1.67348791963502202712110248894, 1.98712168246608165656032518951, 2.72249543771234512317126561289, 3.01466498705365665386079244742, 3.21700579475301496037635134371, 3.52454458944618173453709643275, 3.68409460473347040019768056594, 3.91242906039749612943603470561, 3.98325317501609755883990210461, 4.23980841406719920934813001860, 4.68535793496915786939857257863, 4.86010269480415349883814016780, 5.14063041584431801867210174583, 5.81432259883392504326117803054, 5.99385731499379511746783184476, 6.07699692492010018418819055165, 6.54478385612963626388613185951, 6.61497479796695823445518913407, 6.74991374734458807347112387877, 6.79204294650657684779798179717, 6.79976355877924370065118598333, 7.68933248957775631775006747189
