| L(s) = 1 | − 4-s − 2·5-s − 2·7-s − 2·9-s + 2·11-s + 16-s + 2·20-s + 2·23-s + 3·25-s + 2·28-s + 4·35-s + 2·36-s − 2·43-s − 2·44-s + 4·45-s − 2·47-s + 3·49-s − 4·55-s + 2·61-s + 4·63-s − 2·64-s − 4·77-s − 2·80-s + 3·81-s + 2·83-s − 2·92-s − 4·99-s + ⋯ |
| L(s) = 1 | − 4-s − 2·5-s − 2·7-s − 2·9-s + 2·11-s + 16-s + 2·20-s + 2·23-s + 3·25-s + 2·28-s + 4·35-s + 2·36-s − 2·43-s − 2·44-s + 4·45-s − 2·47-s + 3·49-s − 4·55-s + 2·61-s + 4·63-s − 2·64-s − 4·77-s − 2·80-s + 3·81-s + 2·83-s − 2·92-s − 4·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1025405157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1025405157\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−9.401579418067524432331958793208, −9.261724102180448573909808369698, −9.109533802221884348670597627972, −8.837033921878308270395272134693, −8.603510286675660809996359554799, −8.336196475863634597614208489047, −8.146453088111622490090656980144, −7.88754360637046314271475907853, −7.22636910134097705140984556518, −7.20151037666662089677640548508, −6.88567482910557282915102155226, −6.37343530522119538900024165060, −6.32975189902255865903931757235, −6.29119669582383274707931373042, −5.43205268266639321517703319445, −5.25063307880484960493204905994, −5.06485414514885459340030530931, −4.49483680603590198979315620537, −4.23496781639815852337266858220, −3.63948583156517302384840049644, −3.50118570811506833453859075622, −3.40231692162563963078878105280, −2.99343008038472063889121925162, −2.54179315234813575415517087046, −1.13558389195596783823694363533,
1.13558389195596783823694363533, 2.54179315234813575415517087046, 2.99343008038472063889121925162, 3.40231692162563963078878105280, 3.50118570811506833453859075622, 3.63948583156517302384840049644, 4.23496781639815852337266858220, 4.49483680603590198979315620537, 5.06485414514885459340030530931, 5.25063307880484960493204905994, 5.43205268266639321517703319445, 6.29119669582383274707931373042, 6.32975189902255865903931757235, 6.37343530522119538900024165060, 6.88567482910557282915102155226, 7.20151037666662089677640548508, 7.22636910134097705140984556518, 7.88754360637046314271475907853, 8.146453088111622490090656980144, 8.336196475863634597614208489047, 8.603510286675660809996359554799, 8.837033921878308270395272134693, 9.109533802221884348670597627972, 9.261724102180448573909808369698, 9.401579418067524432331958793208
