| L(s) = 1 | − 32·4-s − 1.08e3·11-s + 768·16-s − 1.18e3·19-s + 1.14e4·29-s − 1.14e4·31-s − 3.08e4·41-s + 3.45e4·44-s − 1.12e4·49-s + 1.01e5·59-s − 5.82e4·61-s − 1.63e4·64-s − 1.23e4·71-s + 3.78e4·76-s − 1.58e4·79-s + 3.29e5·89-s − 1.83e5·101-s − 3.39e5·109-s − 3.64e5·116-s + 3.78e5·121-s + 3.65e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 2.69·11-s + 3/4·16-s − 0.752·19-s + 2.51·29-s − 2.13·31-s − 2.86·41-s + 2.69·44-s − 0.668·49-s + 3.77·59-s − 2.00·61-s − 1/2·64-s − 0.290·71-s + 0.752·76-s − 0.285·79-s + 4.40·89-s − 1.78·101-s − 2.73·109-s − 2.51·116-s + 2.35·121-s + 2.13·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.01630999915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01630999915\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 11230 T^{2} + 229188723 T^{4} + 11230 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 540 T + 248086 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 795290 T^{2} + 317466366123 T^{4} - 795290 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 1787404 T^{2} + 3941614200102 T^{4} + 1787404 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 592 T + 4121589 T^{2} + 592 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 11701772 T^{2} + 67885190172294 T^{4} - 11701772 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5700 T + 48997882 T^{2} - 5700 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5708 T + 64485393 T^{2} + 5708 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 171219020 T^{2} + 14244759565515798 T^{4} - 171219020 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 15420 T + 258100402 T^{2} + 15420 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 444579674 T^{2} + 90169634607517467 T^{4} - 444579674 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 152380028 T^{2} + 88657259104980294 T^{4} - 152380028 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 925638980 T^{2} + 424606861627479798 T^{4} - 925638980 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 50520 T + 1968600982 T^{2} - 50520 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 29126 T + 1603768671 T^{2} + 29126 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 656459306 T^{2} + 3750274361339663307 T^{4} - 656459306 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6180 T + 3034897198 T^{2} + 6180 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 6693286172 T^{2} + 19222787553262718694 T^{4} - 6693286172 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7912 T + 1409684334 T^{2} + 7912 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1661342924 T^{2} + 21907932611565084342 T^{4} - 1661342924 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 164640 T + 13798144114 T^{2} - 164640 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23242017410 T^{2} + \)\(26\!\cdots\!23\)\( T^{4} - 23242017410 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.29217012673260272292628271143, −6.84354385955609948492701883811, −6.68375022076123948206592275496, −6.53319820470295956912767036950, −6.30989184103696967141009195975, −5.54192041591795392700589015443, −5.50756433103629154047430896741, −5.48336996101505174359639949362, −5.36265145295905944149327801824, −4.77495105034567256976420519677, −4.70007357338820478692873572986, −4.41479773886678626797637004549, −4.29292408410368509969550759474, −3.57592797441943129430872061468, −3.51513813819747420924158360374, −3.26937469156100355820515316522, −2.98918303280461391363975354183, −2.43210920727522570763151558566, −2.35976485113464738635970796857, −2.05562221159080450652270936449, −1.63366005478907847984130369464, −1.12478884627645462471873694238, −0.888525669501301943958118625918, −0.33403810327349186413945292587, −0.02873341287027795366089521280,
0.02873341287027795366089521280, 0.33403810327349186413945292587, 0.888525669501301943958118625918, 1.12478884627645462471873694238, 1.63366005478907847984130369464, 2.05562221159080450652270936449, 2.35976485113464738635970796857, 2.43210920727522570763151558566, 2.98918303280461391363975354183, 3.26937469156100355820515316522, 3.51513813819747420924158360374, 3.57592797441943129430872061468, 4.29292408410368509969550759474, 4.41479773886678626797637004549, 4.70007357338820478692873572986, 4.77495105034567256976420519677, 5.36265145295905944149327801824, 5.48336996101505174359639949362, 5.50756433103629154047430896741, 5.54192041591795392700589015443, 6.30989184103696967141009195975, 6.53319820470295956912767036950, 6.68375022076123948206592275496, 6.84354385955609948492701883811, 7.29217012673260272292628271143
