| L(s) = 1 | + 8·2-s + 48·4-s + 46·5-s − 2·7-s + 256·8-s + 368·10-s + 142·11-s − 16·14-s + 1.28e3·16-s + 2.20e3·20-s + 1.13e3·22-s + 625·25-s − 96·28-s + 1.63e3·29-s + 478·31-s + 6.14e3·32-s − 92·35-s + 1.17e4·40-s + 6.81e3·44-s + 2.40e3·49-s + 5.00e3·50-s − 3.21e3·53-s + 6.53e3·55-s − 512·56-s + 1.30e4·58-s − 1.37e4·59-s + 3.82e3·62-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 1.83·5-s − 0.0408·7-s + 4·8-s + 3.67·10-s + 1.17·11-s − 0.0816·14-s + 5·16-s + 5.51·20-s + 2.34·22-s + 25-s − 0.122·28-s + 1.94·29-s + 0.497·31-s + 6·32-s − 0.0751·35-s + 7.35·40-s + 3.52·44-s + 49-s + 2·50-s − 1.14·53-s + 2.15·55-s − 0.163·56-s + 3.89·58-s − 3.94·59-s + 0.994·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(20.57920094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.57920094\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 46 T + 1491 T^{2} - 46 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 2397 T^{2} + 2 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 142 T + 5523 T^{2} - 142 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 818 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 478 T - 695037 T^{2} - 478 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3218 T + 2465043 T^{2} + 3218 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6862 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8158 T + 38154723 T^{2} - 8158 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9118 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4178 T - 30002637 T^{2} + 4178 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17282 T + 210138243 T^{2} + 17282 p^{4} T^{3} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.02951783867068545325985688941, −11.69620266711478493257313606994, −10.96799786217657818756448606848, −10.65236177543510878759151035612, −10.04444948411730475910954174257, −9.690692064818591743031039294994, −9.109228008659545326262349403439, −8.280829843947179134732510646820, −7.66179186552241277189994744719, −6.87707233220923188959162915445, −6.39721403535345579166425181559, −6.25391674741496963488815625597, −5.62422536360054807850068993410, −5.12406153424923022237603183512, −4.42093566933422738312306733703, −3.99147529087032663907428208666, −2.86332146375591210772586427906, −2.71061782259393662520886262847, −1.48914019985824760297896258890, −1.47020803365910127492534811995,
1.47020803365910127492534811995, 1.48914019985824760297896258890, 2.71061782259393662520886262847, 2.86332146375591210772586427906, 3.99147529087032663907428208666, 4.42093566933422738312306733703, 5.12406153424923022237603183512, 5.62422536360054807850068993410, 6.25391674741496963488815625597, 6.39721403535345579166425181559, 6.87707233220923188959162915445, 7.66179186552241277189994744719, 8.280829843947179134732510646820, 9.109228008659545326262349403439, 9.690692064818591743031039294994, 10.04444948411730475910954174257, 10.65236177543510878759151035612, 10.96799786217657818756448606848, 11.69620266711478493257313606994, 12.02951783867068545325985688941
