| L(s) = 1 | + 4-s + 9-s − 4·11-s − 24·31-s + 36-s − 24·41-s − 4·44-s + 40·49-s − 40·59-s − 4·61-s + 56·71-s + 10·89-s − 4·99-s + 76·101-s − 60·109-s − 14·121-s − 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 24·164-s + 167-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 1.20·11-s − 4.31·31-s + 1/6·36-s − 3.74·41-s − 0.603·44-s + 40/7·49-s − 5.20·59-s − 0.512·61-s + 6.64·71-s + 1.05·89-s − 0.402·99-s + 7.56·101-s − 5.74·109-s − 1.27·121-s − 2.15·124-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 − 1.87·164-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9496901045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9496901045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 10 T^{2} + 291 T^{4} - 2300 T^{6} + 64541 T^{8} - 2300 p^{2} T^{10} + 291 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 70 T^{2} + 2451 T^{4} + 57380 T^{6} + 1062941 T^{8} + 57380 p^{2} T^{10} + 2451 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 21 T^{2} + 10 T^{3} + 381 T^{4} + 10 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 10 T^{2} - 429 T^{4} - 9580 T^{6} + 131141 T^{8} - 9580 p^{2} T^{10} - 429 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 834 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 115 T^{2} + 3771 T^{4} - 73735 T^{6} - 7848544 T^{8} - 73735 p^{2} T^{10} + 3771 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 444 p T^{5} + 53 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 160 T^{2} + 10078 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 30 T^{2} + 2331 T^{4} - 108520 T^{6} + 9196101 T^{8} - 108520 p^{2} T^{10} + 2331 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 65 T^{2} - 69 T^{4} - 77395 T^{6} + 12164576 T^{8} - 77395 p^{2} T^{10} - 69 p^{4} T^{12} - 65 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 20 T + 181 T^{2} + 1600 T^{3} + 14601 T^{4} + 1600 p T^{5} + 181 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 170 T^{2} + 6771 T^{4} - 766040 T^{6} - 100558699 T^{8} - 766040 p^{2} T^{10} + 6771 p^{4} T^{12} + 170 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 2126 p T^{5} + 313 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 190 T^{2} + 8931 T^{4} - 1019140 T^{6} - 150091579 T^{8} - 1019140 p^{2} T^{10} + 8931 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 130 T^{2} + 10011 T^{4} + 405860 T^{6} - 16203979 T^{8} + 405860 p^{2} T^{10} + 10011 p^{4} T^{12} + 130 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 5 T - 79 T^{2} + 5 p T^{3} + 5276 T^{4} + 5 p^{2} T^{5} - 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 230 T^{2} + 10731 T^{4} - 1389380 T^{6} - 227949499 T^{8} - 1389380 p^{2} T^{10} + 10731 p^{4} T^{12} + 230 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.48659376697171135755561305992, −4.32749764007327798741658023576, −4.26801703449707484684274055689, −4.18682724529873979598448462326, −4.08259083997038937301626565907, −3.49595768751391087821437953153, −3.46056689213990304059076983353, −3.41935508337437913142117490045, −3.40379260333473074112982901863, −3.39836402484890621659801127265, −3.27173210398008600818189918657, −3.16032632772025399966546773080, −2.62220178118902668420136480152, −2.46620059437453519745052095663, −2.42780217796635724100898570996, −2.18269711460888378422260565610, −2.11542063169241174509695943001, −1.98709266128741669668506756108, −1.91755819944667528095913056520, −1.61094464758277184025631896925, −1.46756618573719022343914165943, −0.990211662751110633057900945061, −0.939225477047680028055061863041, −0.54628984936839894030869685095, −0.14887858938032036473258953315,
0.14887858938032036473258953315, 0.54628984936839894030869685095, 0.939225477047680028055061863041, 0.990211662751110633057900945061, 1.46756618573719022343914165943, 1.61094464758277184025631896925, 1.91755819944667528095913056520, 1.98709266128741669668506756108, 2.11542063169241174509695943001, 2.18269711460888378422260565610, 2.42780217796635724100898570996, 2.46620059437453519745052095663, 2.62220178118902668420136480152, 3.16032632772025399966546773080, 3.27173210398008600818189918657, 3.39836402484890621659801127265, 3.40379260333473074112982901863, 3.41935508337437913142117490045, 3.46056689213990304059076983353, 3.49595768751391087821437953153, 4.08259083997038937301626565907, 4.18682724529873979598448462326, 4.26801703449707484684274055689, 4.32749764007327798741658023576, 4.48659376697171135755561305992
Plot not available for L-functions of degree greater than 10.
