| L(s) = 1 | + 8-s − 18·9-s − 7·64-s − 18·72-s + 189·81-s − 66·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 78·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 0.353·8-s − 6·9-s − 7/8·64-s − 2.12·72-s + 21·81-s − 6·121-s − 0.357·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3528857150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3528857150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{3} + p^{3} T^{6} \) |
| 31 | \( ( 1 + p T^{2} )^{3} \) |
| good | 3 | \( ( 1 + p T^{2} )^{6} \) |
| 5 | \( ( 1 + 2 T^{3} + p^{3} T^{6} )^{2} \) |
| 7 | \( ( 1 - 16 T^{3} + p^{3} T^{6} )( 1 + 16 T^{3} + p^{3} T^{6} ) \) |
| 11 | \( ( 1 + p T^{2} )^{6} \) |
| 13 | \( ( 1 - p T^{2} )^{6} \) |
| 17 | \( ( 1 - p T^{2} )^{6} \) |
| 19 | \( ( 1 - 156 T^{3} + p^{3} T^{6} )( 1 + 156 T^{3} + p^{3} T^{6} ) \) |
| 23 | \( ( 1 + p T^{2} )^{6} \) |
| 29 | \( ( 1 - p T^{2} )^{6} \) |
| 37 | \( ( 1 - p T^{2} )^{6} \) |
| 41 | \( ( 1 + 278 T^{3} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{6} \) |
| 47 | \( ( 1 - 8 T + p T^{2} )^{3}( 1 + 8 T + p T^{2} )^{3} \) |
| 53 | \( ( 1 - p T^{2} )^{6} \) |
| 59 | \( ( 1 - 740 T^{3} + p^{3} T^{6} )( 1 + 740 T^{3} + p^{3} T^{6} ) \) |
| 61 | \( ( 1 - p T^{2} )^{6} \) |
| 67 | \( ( 1 - 12 T + p T^{2} )^{3}( 1 + 12 T + p T^{2} )^{3} \) |
| 71 | \( ( 1 - 1000 T^{3} + p^{3} T^{6} )( 1 + 1000 T^{3} + p^{3} T^{6} ) \) |
| 73 | \( ( 1 - p T^{2} )^{6} \) |
| 79 | \( ( 1 + p T^{2} )^{6} \) |
| 83 | \( ( 1 + p T^{2} )^{6} \) |
| 89 | \( ( 1 - p T^{2} )^{6} \) |
| 97 | \( ( 1 - 1906 T^{3} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73437261808294454190895112147, −7.71128530624617377839738242506, −6.93863920307773152126945015719, −6.80757012245788040692269084572, −6.78325113390828498702889933087, −6.26186825514484817235647295836, −6.24102458107965845536470700985, −6.15084496139736414507903549523, −5.78312674115122735916530698104, −5.61411114973058225388825223302, −5.46131642298374612464632010154, −5.17237416699525419476560873108, −5.09014230766588803961707150480, −5.04862614456172417969031110414, −4.33239507926860768016469451683, −4.24369877966368217687967020678, −3.78960485947151438374689351693, −3.41007972463221342217037733171, −3.35666869492840259854001146037, −2.85162586187524419523999715840, −2.84140508060737160215985701772, −2.55155734352382594554209006106, −2.30348544053450484222815280526, −1.67992813549771985314553470647, −0.44504927924318573743023024872,
0.44504927924318573743023024872, 1.67992813549771985314553470647, 2.30348544053450484222815280526, 2.55155734352382594554209006106, 2.84140508060737160215985701772, 2.85162586187524419523999715840, 3.35666869492840259854001146037, 3.41007972463221342217037733171, 3.78960485947151438374689351693, 4.24369877966368217687967020678, 4.33239507926860768016469451683, 5.04862614456172417969031110414, 5.09014230766588803961707150480, 5.17237416699525419476560873108, 5.46131642298374612464632010154, 5.61411114973058225388825223302, 5.78312674115122735916530698104, 6.15084496139736414507903549523, 6.24102458107965845536470700985, 6.26186825514484817235647295836, 6.78325113390828498702889933087, 6.80757012245788040692269084572, 6.93863920307773152126945015719, 7.71128530624617377839738242506, 7.73437261808294454190895112147
Plot not available for L-functions of degree greater than 10.
