| L(s) = 1 | + 22·5-s − 18·9-s + 148·11-s + 160·19-s + 242·25-s − 100·29-s − 24·31-s + 688·41-s − 396·45-s − 952·49-s + 3.25e3·55-s − 1.33e3·59-s + 488·61-s + 616·71-s − 2.10e3·79-s + 243·81-s − 1.36e3·89-s + 3.52e3·95-s − 2.66e3·99-s + 4.95e3·101-s − 1.10e3·109-s + 8.62e3·121-s + 2.75e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.96·5-s − 2/3·9-s + 4.05·11-s + 1.93·19-s + 1.93·25-s − 0.640·29-s − 0.139·31-s + 2.62·41-s − 1.31·45-s − 2.77·49-s + 7.98·55-s − 2.93·59-s + 1.02·61-s + 1.02·71-s − 2.99·79-s + 1/3·81-s − 1.62·89-s + 3.80·95-s − 2.70·99-s + 4.88·101-s − 0.970·109-s + 6.47·121-s + 1.96·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.900331105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.900331105\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 136 p T^{2} + 457230 T^{4} + 136 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 74 T + 3902 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3728 T^{2} + 11889198 T^{4} - 3728 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10160 T^{2} + 54460638 T^{4} - 10160 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 80 T + 7062 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 31356 T^{2} + 528661862 T^{4} - 31356 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 50 T + 43082 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 17822 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40784 T^{2} + 5454083982 T^{4} - 40784 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 147350 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18964 T^{2} + 2246004198 T^{4} + 18964 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 394272 T^{2} + 83139472430 T^{4} - 394272 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 666 T + 211918 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 424604 T^{2} + 82770213942 T^{4} - 424604 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 308 T + 553262 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 186644 T^{2} - 60009209082 T^{4} - 186644 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1052 T + 1237470 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2190348 T^{2} + 1851255728918 T^{4} - 2190348 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 684 T + 1229686 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1693700 T^{2} + 1935874714758 T^{4} - 1693700 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.459691099489557829666822620245, −9.164387088752878325423512724974, −8.999138419914933678607168032711, −8.905456424066426732818790223438, −8.404399285178460973213610555870, −8.075174455443012467585554249264, −7.52160920096974116156534185047, −7.31455869948494938565759026149, −7.11271283942209307565796333633, −6.43602118167436456770529261870, −6.39164619785762993858077139790, −6.24877512724909560348703093536, −5.94932090656668889734369235222, −5.53345386431883575173458458270, −5.29910222218419789012278081000, −4.74550838166045550077459762798, −4.38897258711049096191606133481, −3.98283864078735772869535659546, −3.62980877620348759504541606460, −3.01376935274297240641932811492, −2.95304287587602149070490726955, −1.98614592785421455403547198093, −1.59749681186365131846113244135, −1.34871222697443959106075745453, −0.78487325745986751004655743653,
0.78487325745986751004655743653, 1.34871222697443959106075745453, 1.59749681186365131846113244135, 1.98614592785421455403547198093, 2.95304287587602149070490726955, 3.01376935274297240641932811492, 3.62980877620348759504541606460, 3.98283864078735772869535659546, 4.38897258711049096191606133481, 4.74550838166045550077459762798, 5.29910222218419789012278081000, 5.53345386431883575173458458270, 5.94932090656668889734369235222, 6.24877512724909560348703093536, 6.39164619785762993858077139790, 6.43602118167436456770529261870, 7.11271283942209307565796333633, 7.31455869948494938565759026149, 7.52160920096974116156534185047, 8.075174455443012467585554249264, 8.404399285178460973213610555870, 8.905456424066426732818790223438, 8.999138419914933678607168032711, 9.164387088752878325423512724974, 9.459691099489557829666822620245
