| L(s) = 1 | + 4·5-s + 96·9-s + 96·11-s + 176·19-s + 56·25-s + 280·29-s + 208·31-s + 360·41-s + 384·45-s − 98·49-s + 384·55-s − 1.13e3·59-s − 936·61-s − 1.94e3·71-s − 1.92e3·79-s + 5.45e3·81-s + 3.30e3·89-s + 704·95-s + 9.21e3·99-s + 2.32e3·101-s − 680·109-s + 2.78e3·121-s + 884·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 32/9·9-s + 2.63·11-s + 2.12·19-s + 0.447·25-s + 1.79·29-s + 1.20·31-s + 1.37·41-s + 1.27·45-s − 2/7·49-s + 0.941·55-s − 2.50·59-s − 1.96·61-s − 3.24·71-s − 2.74·79-s + 7.48·81-s + 3.93·89-s + 0.760·95-s + 9.35·99-s + 2.29·101-s − 0.597·109-s + 2.09·121-s + 0.632·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.05545997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.05545997\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T - 8 p T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 16 p T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 48 T + 2062 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2048 T^{2} + 8213778 T^{4} - 2048 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12348 T^{2} + 83683910 T^{4} - 12348 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 88 T + 15360 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48228 T^{2} + 877537958 T^{4} - 48228 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 140 T + 52502 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 104 T + 61110 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 108716 T^{2} + 6254620278 T^{4} - 108716 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 180 T + 3646 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 135164 T^{2} + 9063402198 T^{4} - 135164 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 354924 T^{2} + 52958993702 T^{4} - 354924 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 138620 T^{2} + 36325958358 T^{4} - 138620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 568 T + 335888 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 468 T + 353192 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 182636 T^{2} + 164284952598 T^{4} - 182636 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 972 T + 950842 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 574892 T^{2} + 285497185350 T^{4} - 574892 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 964 T + 483402 T^{2} + 964 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1072928 T^{2} + 718495235058 T^{4} - 1072928 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1652 T + 1922870 T^{2} - 1652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2626300 T^{2} + 3385412890758 T^{4} - 2626300 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−9.315424929277203759523956589316, −9.013909601365505250258550254869, −8.888655773158358809911859205295, −8.271107270254086428651147049227, −8.034686717511386883039706401409, −7.47724044009935904748725240648, −7.34402724690949211606949213654, −7.20591876931308349376676066968, −7.12760884416461672011807952196, −6.35289163140099936861368961039, −6.28576686029889238037606519772, −6.26904335458791144320364097021, −5.84515140832451865691087190040, −4.95027701724921387320467537940, −4.79065680225589226934253981411, −4.54535518108469429941556528205, −4.39887989727879818704673286221, −3.90981232198982660677045976691, −3.58656283059094135749099961814, −3.13151971107026100110920712185, −2.70348513549447170401699394600, −1.76586993874460737071602861949, −1.45652072636287962982940390657, −1.10895177161931539548031696215, −0.999589175542695498118222888588,
0.999589175542695498118222888588, 1.10895177161931539548031696215, 1.45652072636287962982940390657, 1.76586993874460737071602861949, 2.70348513549447170401699394600, 3.13151971107026100110920712185, 3.58656283059094135749099961814, 3.90981232198982660677045976691, 4.39887989727879818704673286221, 4.54535518108469429941556528205, 4.79065680225589226934253981411, 4.95027701724921387320467537940, 5.84515140832451865691087190040, 6.26904335458791144320364097021, 6.28576686029889238037606519772, 6.35289163140099936861368961039, 7.12760884416461672011807952196, 7.20591876931308349376676066968, 7.34402724690949211606949213654, 7.47724044009935904748725240648, 8.034686717511386883039706401409, 8.271107270254086428651147049227, 8.888655773158358809911859205295, 9.013909601365505250258550254869, 9.315424929277203759523956589316
