| L(s) = 1 | − 2·9-s − 2·16-s + 3·81-s + 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2·16-s + 3·81-s + 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7298669652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7298669652\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.09560513501837712300660610237, −6.05327040200752239837873592710, −6.01315399648944472947344957014, −5.71697972405204197144825293321, −5.50770477048211774475876571931, −5.11073602558918721954878506660, −4.91941276626149735131528971085, −4.86299519466543540237280624927, −4.73911736337185153559786748594, −4.51571736393598425069584229811, −4.20988410176231920015862645776, −4.03948349753073161448589278225, −3.68044538549805161037194842991, −3.43786074872914155191521368924, −3.29225496567429635124775422546, −3.13190296037312137611406742292, −2.99808214562133670811835669967, −2.44302775431214962710785042219, −2.28217696657116332585095021222, −2.25159638984118446757458155563, −2.06566702077787859214345016524, −1.69013764664760250571454696779, −1.18484180527247312075975038603, −0.822940456763734131056439174555, −0.37246897971375030824601561153,
0.37246897971375030824601561153, 0.822940456763734131056439174555, 1.18484180527247312075975038603, 1.69013764664760250571454696779, 2.06566702077787859214345016524, 2.25159638984118446757458155563, 2.28217696657116332585095021222, 2.44302775431214962710785042219, 2.99808214562133670811835669967, 3.13190296037312137611406742292, 3.29225496567429635124775422546, 3.43786074872914155191521368924, 3.68044538549805161037194842991, 4.03948349753073161448589278225, 4.20988410176231920015862645776, 4.51571736393598425069584229811, 4.73911736337185153559786748594, 4.86299519466543540237280624927, 4.91941276626149735131528971085, 5.11073602558918721954878506660, 5.50770477048211774475876571931, 5.71697972405204197144825293321, 6.01315399648944472947344957014, 6.05327040200752239837873592710, 6.09560513501837712300660610237
