| L(s) = 1 | + 2·4-s − 6·11-s + 16-s − 4·19-s + 6·29-s + 4·31-s − 12·41-s − 12·44-s − 11·49-s − 6·59-s − 2·61-s − 2·64-s + 48·71-s − 8·76-s + 8·79-s + 60·89-s − 12·101-s − 100·109-s + 12·116-s + 41·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 1.80·11-s + 1/4·16-s − 0.917·19-s + 1.11·29-s + 0.718·31-s − 1.87·41-s − 1.80·44-s − 1.57·49-s − 0.781·59-s − 0.256·61-s − 1/4·64-s + 5.69·71-s − 0.917·76-s + 0.900·79-s + 6.35·89-s − 1.19·101-s − 9.57·109-s + 1.11·116-s + 3.72·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.589298855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.589298855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 11 T^{2} + T^{4} + 242 T^{6} + 6070 T^{8} + 242 p^{2} T^{10} + p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 16 T^{2} - 14 T^{4} + 1088 T^{6} + 1075 T^{8} + 1088 p^{2} T^{10} - 14 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 11 T^{2} - 60 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 84206 p^{2} T^{10} + 2797 p^{4} T^{12} + 71 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 11 T^{2} - 1223 T^{4} - 25894 T^{6} - 1836194 T^{8} - 25894 p^{2} T^{10} - 1223 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 131 T^{2} + 9121 T^{4} + 474482 T^{6} + 21853270 T^{8} + 474482 p^{2} T^{10} + 9121 p^{4} T^{12} + 131 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 18 p T^{5} - 103 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 83 T^{2} + 3396 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 275 T^{2} + 43609 T^{4} + 5015450 T^{6} + 456585310 T^{8} + 5015450 p^{2} T^{10} + 43609 p^{4} T^{12} + 275 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 311 T^{2} + 54721 T^{4} + 7209602 T^{6} + 765422830 T^{8} + 7209602 p^{2} T^{10} + 54721 p^{4} T^{12} + 311 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.20196542038691277560214381718, −3.79771100643049636755936628374, −3.75821179395223401411471501517, −3.74216918410172504472015567338, −3.57511138603100176293555448790, −3.43393777991605547843324304313, −3.24721073295966713832437228983, −3.16390162769281882420786764955, −2.99881717133400595590332070085, −2.81727445299716792283140930094, −2.72389218972973398508018893384, −2.61362542045687639282899471636, −2.56204006399005974159568528600, −2.36834692083162220813708264346, −2.15519019057184065534929319212, −1.95682745269629563300845312724, −1.87979702231278282581417475944, −1.87143756316764391814557276019, −1.58894182445273535490185742261, −1.42456103029138621281479057331, −1.23346795810376610003134818824, −0.78761058339001017855222096227, −0.67972239734047404614014648393, −0.42388302104703337868282555766, −0.38037000580233207972053987786,
0.38037000580233207972053987786, 0.42388302104703337868282555766, 0.67972239734047404614014648393, 0.78761058339001017855222096227, 1.23346795810376610003134818824, 1.42456103029138621281479057331, 1.58894182445273535490185742261, 1.87143756316764391814557276019, 1.87979702231278282581417475944, 1.95682745269629563300845312724, 2.15519019057184065534929319212, 2.36834692083162220813708264346, 2.56204006399005974159568528600, 2.61362542045687639282899471636, 2.72389218972973398508018893384, 2.81727445299716792283140930094, 2.99881717133400595590332070085, 3.16390162769281882420786764955, 3.24721073295966713832437228983, 3.43393777991605547843324304313, 3.57511138603100176293555448790, 3.74216918410172504472015567338, 3.75821179395223401411471501517, 3.79771100643049636755936628374, 4.20196542038691277560214381718
Plot not available for L-functions of degree greater than 10.
