| L(s) = 1 | + 5·4-s + 100·7-s + 508·13-s − 231·16-s + 502·19-s + 575·25-s + 500·28-s + 760·31-s − 2.64e3·37-s − 7.17e3·43-s + 2.69e3·49-s + 2.54e3·52-s + 3.65e3·61-s − 2.43e3·64-s + 1.04e4·67-s − 6.06e3·73-s + 2.51e3·76-s − 8.72e3·79-s + 5.08e4·91-s − 638·97-s + 2.87e3·100-s − 5.40e3·103-s + 3.57e4·109-s − 2.31e4·112-s + 2.88e4·121-s + 3.80e3·124-s + 127-s + ⋯ |
| L(s) = 1 | + 5/16·4-s + 2.04·7-s + 3.00·13-s − 0.902·16-s + 1.39·19-s + 0.919·25-s + 0.637·28-s + 0.790·31-s − 1.93·37-s − 3.87·43-s + 1.12·49-s + 0.939·52-s + 0.981·61-s − 0.594·64-s + 2.32·67-s − 1.13·73-s + 0.434·76-s − 1.39·79-s + 6.13·91-s − 0.0678·97-s + 0.287·100-s − 0.509·103-s + 3.00·109-s − 1.84·112-s + 1.97·121-s + 0.247·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.240483546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.240483546\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 23 p^{2} T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 50 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 28850 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 254 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 49430 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 251 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 445607 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 235175 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 380 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1324 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3939310 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3586 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7805615 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9570985 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 21002390 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1826 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5225 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 37533625 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3031 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4360 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 49286642 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 16401782 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 319 T + p^{4} T^{2} )^{2} \) |
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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
−11.56009060675846899974405245671, −11.14759179014839746583334287455, −11.13675647166065275432012171152, −10.27843271156613769252795823476, −9.985179102911996214008946711098, −9.001295762099844554564730889076, −8.513982067008773452064136804619, −8.415640134365543247855572382525, −8.008685896119148202822088101819, −7.03517456156328152772787440703, −6.81054912841957016498105491925, −6.13019691255950638949609676522, −5.38521187691829696308409834508, −5.00701714878132969901250954231, −4.43512215300582690988699328069, −3.53209867055230237130081352264, −3.18400682468566225901821312266, −1.75974861486893820010017372984, −1.60978348121035150337990537983, −0.825624583136896343668328296444,
0.825624583136896343668328296444, 1.60978348121035150337990537983, 1.75974861486893820010017372984, 3.18400682468566225901821312266, 3.53209867055230237130081352264, 4.43512215300582690988699328069, 5.00701714878132969901250954231, 5.38521187691829696308409834508, 6.13019691255950638949609676522, 6.81054912841957016498105491925, 7.03517456156328152772787440703, 8.008685896119148202822088101819, 8.415640134365543247855572382525, 8.513982067008773452064136804619, 9.001295762099844554564730889076, 9.985179102911996214008946711098, 10.27843271156613769252795823476, 11.13675647166065275432012171152, 11.14759179014839746583334287455, 11.56009060675846899974405245671
