| L(s) = 1 | + 8·4-s + 24·16-s + 12·19-s + 4·25-s + 36·31-s − 14·49-s − 72·61-s + 96·76-s + 28·79-s + 32·100-s + 28·109-s − 28·121-s + 288·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 76·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4·4-s + 6·16-s + 2.75·19-s + 4/5·25-s + 6.46·31-s − 2·49-s − 9.21·61-s + 11.0·76-s + 3.15·79-s + 16/5·100-s + 2.68·109-s − 2.54·121-s + 25.8·124-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.21988158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.21988158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 20 T^{2} + 111 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 53 T^{2} + 1440 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 80 T^{2} + 4191 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 62 T^{2} + 1035 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 64 T^{2} + 615 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 18 T + 169 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 55 T^{2} - 1464 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 116 T^{2} + 5535 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−5.34106369036749191927257592183, −4.85813751194261999648974079246, −4.80508915434941054317740606218, −4.74281453196294598390763398854, −4.69387871553803143725041862184, −4.57227526069810974224825099700, −4.54355625121611571988546222884, −4.19817289567899255163706148673, −4.14759282954120913626160549220, −3.48409080750250697950495189354, −3.47244974783512346868170146401, −3.34185022527296286517700194107, −3.14272419196782943660328963400, −3.13441953674736224493685892957, −2.78331795164472451717090859991, −2.76505091599599628999474194035, −2.67935021227277862327155334990, −2.55856420283690482827404677128, −2.15153089058037926028135939291, −2.10909762327495766017632077216, −1.72351783547206927813873004332, −1.35644189542647293346596156062, −1.34153190557127302939483608156, −1.33329232229953231157524545617, −0.62627639910159928857147692819,
0.62627639910159928857147692819, 1.33329232229953231157524545617, 1.34153190557127302939483608156, 1.35644189542647293346596156062, 1.72351783547206927813873004332, 2.10909762327495766017632077216, 2.15153089058037926028135939291, 2.55856420283690482827404677128, 2.67935021227277862327155334990, 2.76505091599599628999474194035, 2.78331795164472451717090859991, 3.13441953674736224493685892957, 3.14272419196782943660328963400, 3.34185022527296286517700194107, 3.47244974783512346868170146401, 3.48409080750250697950495189354, 4.14759282954120913626160549220, 4.19817289567899255163706148673, 4.54355625121611571988546222884, 4.57227526069810974224825099700, 4.69387871553803143725041862184, 4.74281453196294598390763398854, 4.80508915434941054317740606218, 4.85813751194261999648974079246, 5.34106369036749191927257592183
Plot not available for L-functions of degree greater than 10.
