| L(s) = 1 | − 6·7-s + 14·13-s − 38·19-s + 42·25-s − 20·31-s + 126·37-s − 100·43-s − 71·49-s + 158·61-s + 154·67-s − 34·73-s − 22·79-s − 84·91-s − 194·97-s + 266·103-s + 260·109-s − 150·121-s + 127-s + 131-s + 228·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 6/7·7-s + 1.07·13-s − 2·19-s + 1.67·25-s − 0.645·31-s + 3.40·37-s − 2.32·43-s − 1.44·49-s + 2.59·61-s + 2.29·67-s − 0.465·73-s − 0.278·79-s − 0.923·91-s − 2·97-s + 2.58·103-s + 2.38·109-s − 1.23·121-s + 0.00787·127-s + 0.00763·131-s + 12/7·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.592044277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592044277\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 42 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 150 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 378 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1050 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 910 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 63 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2618 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2838 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 79 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 77 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3810 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12210 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14042 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
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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
−12.69305762787801526711545289047, −11.70977860610617227098567862651, −11.22638498827461634904094621148, −11.04506956382298680623223183801, −10.39482482914812763809836030204, −9.825943646897817915613913547795, −9.539435407952949826363109546005, −8.774299524182894108602763993439, −8.359929886661361973947746881358, −8.101326986853750701934163540746, −7.09492117274099749645533492537, −6.53007635444527891597178741168, −6.40110719362524553800153061383, −5.67992556246414970759367312034, −4.86084522784574180344487689848, −4.24801590395536063941728662376, −3.57282744992988254813882842688, −2.88492772039715598952188767432, −1.98666229873596286392538499627, −0.71148437845598326309032742967,
0.71148437845598326309032742967, 1.98666229873596286392538499627, 2.88492772039715598952188767432, 3.57282744992988254813882842688, 4.24801590395536063941728662376, 4.86084522784574180344487689848, 5.67992556246414970759367312034, 6.40110719362524553800153061383, 6.53007635444527891597178741168, 7.09492117274099749645533492537, 8.101326986853750701934163540746, 8.359929886661361973947746881358, 8.774299524182894108602763993439, 9.539435407952949826363109546005, 9.825943646897817915613913547795, 10.39482482914812763809836030204, 11.04506956382298680623223183801, 11.22638498827461634904094621148, 11.70977860610617227098567862651, 12.69305762787801526711545289047
