| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1869900853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1869900853\) |
| \(L(\frac{5}{2})\) |
|
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
−6.82173474593088831284943526072, −6.76758807636533982879548005677, −6.48523255203081336021150763756, −5.86423273286072830583365177404, −5.85190979733750193427746935887, −5.51080071241078138719220741387, −5.48109088318667325212095964016, −5.18663533659423481190396093977, −4.98020717173412635681943499803, −4.49852494971327762959357801024, −4.42111742583327510673412378804, −4.22197117415714484262611280425, −4.15979795246281205144207086746, −3.60854358608341440167908466835, −3.37492841695897209515294619006, −2.95578159069212447572066193867, −2.83315352689586680227219239077, −2.67808413607816863972215345814, −2.46070596901356179618209060246, −2.08137921179789906170807221378, −1.74797225225809062209640747076, −1.15710929027036725883454087288, −0.69979004844029313231538163536, −0.23820130380906611051906225258, −0.18227004539074253846466594989,
0.18227004539074253846466594989, 0.23820130380906611051906225258, 0.69979004844029313231538163536, 1.15710929027036725883454087288, 1.74797225225809062209640747076, 2.08137921179789906170807221378, 2.46070596901356179618209060246, 2.67808413607816863972215345814, 2.83315352689586680227219239077, 2.95578159069212447572066193867, 3.37492841695897209515294619006, 3.60854358608341440167908466835, 4.15979795246281205144207086746, 4.22197117415714484262611280425, 4.42111742583327510673412378804, 4.49852494971327762959357801024, 4.98020717173412635681943499803, 5.18663533659423481190396093977, 5.48109088318667325212095964016, 5.51080071241078138719220741387, 5.85190979733750193427746935887, 5.86423273286072830583365177404, 6.48523255203081336021150763756, 6.76758807636533982879548005677, 6.82173474593088831284943526072
