| L(s) = 1 | − 4·4-s − 12·11-s + 9·16-s − 8·19-s − 2·25-s + 12·29-s + 4·31-s − 24·41-s + 48·44-s − 16·49-s + 24·59-s − 8·61-s − 12·64-s + 72·71-s + 32·76-s + 16·79-s + 9·81-s − 24·89-s + 8·100-s − 24·101-s − 32·109-s − 48·116-s + 92·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.61·11-s + 9/4·16-s − 1.83·19-s − 2/5·25-s + 2.22·29-s + 0.718·31-s − 3.74·41-s + 7.23·44-s − 2.28·49-s + 3.12·59-s − 1.02·61-s − 3/2·64-s + 8.54·71-s + 3.67·76-s + 1.80·79-s + 81-s − 2.54·89-s + 4/5·100-s − 2.38·101-s − 3.06·109-s − 4.45·116-s + 8.36·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1163243857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1163243857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 2 | \( 1 + p^{2} T^{2} + 7 T^{4} + p^{2} T^{6} + T^{8} + p^{4} T^{10} + 7 p^{4} T^{12} + p^{8} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 16 T^{2} + 121 T^{4} + 592 T^{6} + 3280 T^{8} + 592 p^{2} T^{10} + 121 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 88 T^{2} + 4753 T^{4} + 170104 T^{6} + 4569664 T^{8} + 170104 p^{2} T^{10} + 4753 p^{4} T^{12} + 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 4 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 12 T + 29 T^{2} + 396 T^{3} + 5640 T^{4} + 396 p T^{5} + 29 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 88 T^{2} + 3838 T^{4} + 18304 T^{6} - 2480621 T^{8} + 18304 p^{2} T^{10} + 3838 p^{4} T^{12} + 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 160 T^{2} + 14809 T^{4} + 1019680 T^{6} + 55015600 T^{8} + 1019680 p^{2} T^{10} + 14809 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 196 T^{2} + 15174 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 12 T + 2 T^{2} - 288 T^{3} + 8187 T^{4} - 288 p T^{5} + 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 284 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 184 T^{2} + 16657 T^{4} + 1512664 T^{6} + 123674896 T^{8} + 1512664 p^{2} T^{10} + 16657 p^{4} T^{12} + 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 8 T - 98 T^{2} - 32 T^{3} + 14947 T^{4} - 32 p T^{5} - 98 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 160 T^{2} + 5929 T^{4} + 11360 p T^{6} + 23680 p^{2} T^{8} + 11360 p^{3} T^{10} + 5929 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 52 T^{2} - 12902 T^{4} - 167024 T^{6} + 129576019 T^{8} - 167024 p^{2} T^{10} - 12902 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−7.951088943996130534695981124934, −7.88515502509424774423259794821, −7.55391847213597745140673027120, −7.51889093204330101270140991174, −6.95480278415243523552271718165, −6.77799752033855071011296975596, −6.67073109823986074166042756097, −6.49555265506791225858237422666, −6.34447043357801617997453194865, −6.29060068472210823633506581082, −5.53845487395368494958012742528, −5.36138558065511476615761428426, −5.29860820502979910190626681338, −5.26400770260162682424791785396, −5.05925354367494962075770773831, −4.82087524989054237548181658641, −4.67693449825763920894240738022, −4.29990823860780370590205816847, −3.93576124535336025159435104885, −3.79044380172160831923546015431, −3.53658687071640279818941773544, −3.05579008791127064836533074837, −2.80491962154153509206262157101, −2.26244947342011222834222574810, −2.20090208401029123785420405987,
2.20090208401029123785420405987, 2.26244947342011222834222574810, 2.80491962154153509206262157101, 3.05579008791127064836533074837, 3.53658687071640279818941773544, 3.79044380172160831923546015431, 3.93576124535336025159435104885, 4.29990823860780370590205816847, 4.67693449825763920894240738022, 4.82087524989054237548181658641, 5.05925354367494962075770773831, 5.26400770260162682424791785396, 5.29860820502979910190626681338, 5.36138558065511476615761428426, 5.53845487395368494958012742528, 6.29060068472210823633506581082, 6.34447043357801617997453194865, 6.49555265506791225858237422666, 6.67073109823986074166042756097, 6.77799752033855071011296975596, 6.95480278415243523552271718165, 7.51889093204330101270140991174, 7.55391847213597745140673027120, 7.88515502509424774423259794821, 7.951088943996130534695981124934
Plot not available for L-functions of degree greater than 10.
