| L(s) = 1 | + 14·4-s − 56·7-s − 224·13-s − 60·16-s + 1.12e3·19-s + 368·25-s − 784·28-s − 728·31-s − 1.65e3·37-s + 3.47e3·43-s − 2.45e3·49-s − 3.13e3·52-s + 5.23e3·61-s − 4.42e3·64-s − 7.56e3·67-s + 1.32e4·73-s + 1.56e4·76-s − 8.55e3·79-s + 1.25e4·91-s − 1.16e4·97-s + 5.15e3·100-s − 2.99e4·103-s + 2.12e4·109-s + 3.36e3·112-s + 2.89e4·121-s − 1.01e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 8/7·7-s − 1.32·13-s − 0.234·16-s + 3.10·19-s + 0.588·25-s − 28-s − 0.757·31-s − 1.20·37-s + 1.87·43-s − 1.02·49-s − 1.15·52-s + 1.40·61-s − 1.08·64-s − 1.68·67-s + 2.48·73-s + 2.71·76-s − 1.37·79-s + 1.51·91-s − 1.23·97-s + 0.515·100-s − 2.82·103-s + 1.79·109-s + 0.267·112-s + 1.98·121-s − 0.662·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.043249471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043249471\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 p T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 368 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 p T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 28994 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 112 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 159104 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 560 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 76510 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 437360 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 364 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 826 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2369600 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1736 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8051810 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12560240 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3856994 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2618 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3784 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 23206750 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6608 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4276 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 94902530 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 106425344 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5824 T + p^{4} T^{2} )^{2} \) |
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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
−20.79834550994884979452667231040, −20.39814851515352733791280225936, −19.68945706363571796124802158745, −19.23754369912055127718612017883, −18.24221438119862436751232998949, −17.56178094444797750077449670346, −16.45086328334429934111289160151, −16.11414509296955556598322522095, −15.51072000936153803521639759887, −14.43674555261655475358626398397, −13.69219155187576034968971006764, −12.58650245369950066745516405039, −11.97386217800563039958632606030, −11.08512464056210486461730968623, −9.880286965419803764732677726567, −9.324655210416613825846300216378, −7.50214236652905414179581649637, −6.87871834439669389065105973666, −5.39176983631017012344750806467, −3.02878905212755916404233430378,
3.02878905212755916404233430378, 5.39176983631017012344750806467, 6.87871834439669389065105973666, 7.50214236652905414179581649637, 9.324655210416613825846300216378, 9.880286965419803764732677726567, 11.08512464056210486461730968623, 11.97386217800563039958632606030, 12.58650245369950066745516405039, 13.69219155187576034968971006764, 14.43674555261655475358626398397, 15.51072000936153803521639759887, 16.11414509296955556598322522095, 16.45086328334429934111289160151, 17.56178094444797750077449670346, 18.24221438119862436751232998949, 19.23754369912055127718612017883, 19.68945706363571796124802158745, 20.39814851515352733791280225936, 20.79834550994884979452667231040
