| L(s) = 1 | − 5·4-s + 2·13-s + 9·16-s + 20·19-s − 19·25-s + 10·31-s + 18·37-s + 28·43-s − 26·49-s − 10·52-s + 54·61-s − 10·64-s + 38·67-s − 16·73-s − 100·76-s + 14·79-s − 6·97-s + 95·100-s − 14·103-s + 2·109-s − 34·121-s − 50·124-s + 127-s + 131-s + 137-s + 139-s − 90·148-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 0.554·13-s + 9/4·16-s + 4.58·19-s − 3.79·25-s + 1.79·31-s + 2.95·37-s + 4.26·43-s − 3.71·49-s − 1.38·52-s + 6.91·61-s − 5/4·64-s + 4.64·67-s − 1.87·73-s − 11.4·76-s + 1.57·79-s − 0.609·97-s + 19/2·100-s − 1.37·103-s + 0.191·109-s − 3.09·121-s − 4.49·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.39·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 29^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.52657253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.52657253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 5 T^{2} + p^{4} T^{4} + 45 T^{6} + 101 T^{8} + 45 p^{2} T^{10} + p^{8} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + 19 T^{2} + 191 T^{4} + 1429 T^{6} + 8256 T^{8} + 1429 p^{2} T^{10} + 191 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 13 T^{2} + 20 T^{3} + 89 T^{4} + 20 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 34 T^{2} + 645 T^{4} + 8726 T^{6} + 9004 p T^{8} + 8726 p^{2} T^{10} + 645 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - T + 38 T^{2} - 30 T^{3} + 651 T^{4} - 30 p T^{5} + 38 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 52 T^{2} + 1460 T^{4} + 30092 T^{6} + 527254 T^{8} + 30092 p^{2} T^{10} + 1460 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 5 T + 43 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 35 T^{2} + 1571 T^{4} + 39685 T^{6} + 1089016 T^{8} + 39685 p^{2} T^{10} + 1571 p^{4} T^{12} + 35 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 5 T + 114 T^{2} - 460 T^{3} + 5151 T^{4} - 460 p T^{5} + 114 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 9 T + 109 T^{2} - 673 T^{3} + 5324 T^{4} - 673 p T^{5} + 109 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 162 T^{2} + 14413 T^{4} + 889934 T^{6} + 41235380 T^{8} + 889934 p^{2} T^{10} + 14413 p^{4} T^{12} + 162 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 14 T + 103 T^{2} - 960 T^{3} + 8201 T^{4} - 960 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 155 T^{2} + 12331 T^{4} + 782445 T^{6} + 41499176 T^{8} + 782445 p^{2} T^{10} + 12331 p^{4} T^{12} + 155 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 144 T^{2} + 10622 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 211 T^{2} + 24975 T^{4} + 2115569 T^{6} + 138442424 T^{8} + 2115569 p^{2} T^{10} + 24975 p^{4} T^{12} + 211 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 27 T + 498 T^{2} - 5874 T^{3} + 54255 T^{4} - 5874 p T^{5} + 498 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 19 T + 304 T^{2} - 2838 T^{3} + 27239 T^{4} - 2838 p T^{5} + 304 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 103 T^{2} + 17543 T^{4} + 1340981 T^{6} + 130159480 T^{8} + 1340981 p^{2} T^{10} + 17543 p^{4} T^{12} + 103 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 8 T + 131 T^{2} + 1094 T^{3} + 14619 T^{4} + 1094 p T^{5} + 131 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 7 T + 220 T^{2} - 1272 T^{3} + 24269 T^{4} - 1272 p T^{5} + 220 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 138 T^{2} + 19885 T^{4} + 1799198 T^{6} + 178905044 T^{8} + 1799198 p^{2} T^{10} + 19885 p^{4} T^{12} + 138 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 438 T^{2} + 93973 T^{4} + 13385366 T^{6} + 1386133340 T^{8} + 13385366 p^{2} T^{10} + 93973 p^{4} T^{12} + 438 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 3 T + 167 T^{2} + 5 p T^{3} + 18096 T^{4} + 5 p^{2} T^{5} + 167 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.25587017742218836587592774088, −3.01436607424833740306909198472, −2.98456094925968581543986238385, −2.94787983822079500233073119316, −2.78854821328600252291656389107, −2.65278016016051110062197805639, −2.49313691594091675791613929423, −2.43831319009164804275880040193, −2.35267056768649447646175877615, −2.29529959927873990136212201101, −2.23701910420667289000917495422, −1.79545310967620180218745117614, −1.70924272134575210928263780891, −1.68973857248096997756552697184, −1.67368061418384315615505288318, −1.52832090849494420631448839289, −1.32718891362557278936359078297, −1.00693681199066280822759296783, −0.891200533339698870697225726519, −0.797915353672243560754955172632, −0.74042989821156927320148775086, −0.61915879484559102634809750169, −0.54797940048731320516837461728, −0.51927517703419244216967404636, −0.24122970998436824536408134927,
0.24122970998436824536408134927, 0.51927517703419244216967404636, 0.54797940048731320516837461728, 0.61915879484559102634809750169, 0.74042989821156927320148775086, 0.797915353672243560754955172632, 0.891200533339698870697225726519, 1.00693681199066280822759296783, 1.32718891362557278936359078297, 1.52832090849494420631448839289, 1.67368061418384315615505288318, 1.68973857248096997756552697184, 1.70924272134575210928263780891, 1.79545310967620180218745117614, 2.23701910420667289000917495422, 2.29529959927873990136212201101, 2.35267056768649447646175877615, 2.43831319009164804275880040193, 2.49313691594091675791613929423, 2.65278016016051110062197805639, 2.78854821328600252291656389107, 2.94787983822079500233073119316, 2.98456094925968581543986238385, 3.01436607424833740306909198472, 3.25587017742218836587592774088
Plot not available for L-functions of degree greater than 10.
