| L(s) = 1 | − 4·2-s − 3·3-s + 6·4-s + 4·5-s + 12·6-s + 4·7-s + 6·9-s − 16·10-s + 2·11-s − 18·12-s + 3·13-s − 16·14-s − 12·15-s − 15·16-s + 4·17-s − 24·18-s + 6·19-s + 24·20-s − 12·21-s − 8·22-s − 6·23-s + 6·25-s − 12·26-s − 3·27-s + 24·28-s − 12·29-s + 48·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.73·3-s + 3·4-s + 1.78·5-s + 4.89·6-s + 1.51·7-s + 2·9-s − 5.05·10-s + 0.603·11-s − 5.19·12-s + 0.832·13-s − 4.27·14-s − 3.09·15-s − 3.75·16-s + 0.970·17-s − 5.65·18-s + 1.37·19-s + 5.36·20-s − 2.61·21-s − 1.70·22-s − 1.25·23-s + 6/5·25-s − 2.35·26-s − 0.577·27-s + 4.53·28-s − 2.22·29-s + 8.76·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7000216464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7000216464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( 1 + p T + p T^{2} - 2 p T^{3} - 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( ( 1 - T + T^{2} )^{4} \) |
| good | 11 | \( 1 - 2 T - 10 T^{2} + 96 T^{3} - 133 T^{4} - 730 T^{5} + 2998 T^{6} + 336 T^{7} - 27968 T^{8} + 336 p T^{9} + 2998 p^{2} T^{10} - 730 p^{3} T^{11} - 133 p^{4} T^{12} + 96 p^{5} T^{13} - 10 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 3 T - 31 T^{2} + 42 T^{3} + 673 T^{4} - 189 T^{5} - 10438 T^{6} + 1965 T^{7} + 123148 T^{8} + 1965 p T^{9} - 10438 p^{2} T^{10} - 189 p^{3} T^{11} + 673 p^{4} T^{12} + 42 p^{5} T^{13} - 31 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 8 T^{2} + 40 T^{3} + 37 T^{4} + 40 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3 T + 37 T^{2} - 108 T^{3} + 1098 T^{4} - 108 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 6 T - 38 T^{2} - 12 p T^{3} + 862 T^{4} + 5472 T^{5} - 23384 T^{6} - 34122 T^{7} + 740695 T^{8} - 34122 p T^{9} - 23384 p^{2} T^{10} + 5472 p^{3} T^{11} + 862 p^{4} T^{12} - 12 p^{6} T^{13} - 38 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 12 T + 82 T^{2} + 60 T^{3} - 2498 T^{4} - 22590 T^{5} - 36524 T^{6} + 453936 T^{7} + 4700635 T^{8} + 453936 p T^{9} - 36524 p^{2} T^{10} - 22590 p^{3} T^{11} - 2498 p^{4} T^{12} + 60 p^{5} T^{13} + 82 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 + 2 T + 48 T^{2} - 140 T^{3} - 442 T^{4} - 9642 T^{5} + 16840 T^{6} + 106418 T^{7} + 2843631 T^{8} + 106418 p T^{9} + 16840 p^{2} T^{10} - 9642 p^{3} T^{11} - 442 p^{4} T^{12} - 140 p^{5} T^{13} + 48 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 4 T + 64 T^{2} + 322 T^{3} + 2962 T^{4} + 322 p T^{5} + 64 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - T - 148 T^{2} + 111 T^{3} + 13211 T^{4} - 6980 T^{5} - 807632 T^{6} + 117150 T^{7} + 37798900 T^{8} + 117150 p T^{9} - 807632 p^{2} T^{10} - 6980 p^{3} T^{11} + 13211 p^{4} T^{12} + 111 p^{5} T^{13} - 148 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 5 T - 96 T^{2} + 203 T^{3} + 6131 T^{4} - 1356 T^{5} - 261476 T^{6} + 65884 T^{7} + 8587152 T^{8} + 65884 p T^{9} - 261476 p^{2} T^{10} - 1356 p^{3} T^{11} + 6131 p^{4} T^{12} + 203 p^{5} T^{13} - 96 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 + 11 T - T^{2} - 600 T^{3} - 3319 T^{4} - 1385 T^{5} + 62146 T^{6} + 274965 T^{7} - 366314 T^{8} + 274965 p T^{9} + 62146 p^{2} T^{10} - 1385 p^{3} T^{11} - 3319 p^{4} T^{12} - 600 p^{5} T^{13} - p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 22 T + 368 T^{2} - 3856 T^{3} + 33406 T^{4} - 3856 p T^{5} + 368 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + T - 94 T^{2} + 879 T^{3} + 4265 T^{4} - 70672 T^{5} + 348742 T^{6} + 2910810 T^{7} - 25645700 T^{8} + 2910810 p T^{9} + 348742 p^{2} T^{10} - 70672 p^{3} T^{11} + 4265 p^{4} T^{12} + 879 p^{5} T^{13} - 94 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 4 T - 132 T^{2} - 1084 T^{3} + 7670 T^{4} + 88086 T^{5} - 11348 T^{6} - 2901056 T^{7} - 12506193 T^{8} - 2901056 p T^{9} - 11348 p^{2} T^{10} + 88086 p^{3} T^{11} + 7670 p^{4} T^{12} - 1084 p^{5} T^{13} - 132 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 21 T + 92 T^{2} - 519 T^{3} + 1771 T^{4} + 52092 T^{5} - 121444 T^{6} + 3350544 T^{7} + 82422040 T^{8} + 3350544 p T^{9} - 121444 p^{2} T^{10} + 52092 p^{3} T^{11} + 1771 p^{4} T^{12} - 519 p^{5} T^{13} + 92 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 17 T + 362 T^{2} - 3707 T^{3} + 41248 T^{4} - 3707 p T^{5} + 362 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 10 T + 268 T^{2} + 1828 T^{3} + 27853 T^{4} + 1828 p T^{5} + 268 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 25 T + 135 T^{2} + 152 T^{3} + 25385 T^{4} + 267381 T^{5} - 350294 T^{6} + 4515655 T^{7} + 219222990 T^{8} + 4515655 p T^{9} - 350294 p^{2} T^{10} + 267381 p^{3} T^{11} + 25385 p^{4} T^{12} + 152 p^{5} T^{13} + 135 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 + 23 T + 209 T^{2} + 1890 T^{3} + 15209 T^{4} - 14951 T^{5} - 1103450 T^{6} - 15006849 T^{7} - 178990154 T^{8} - 15006849 p T^{9} - 1103450 p^{2} T^{10} - 14951 p^{3} T^{11} + 15209 p^{4} T^{12} + 1890 p^{5} T^{13} + 209 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 16 T + 332 T^{2} - 3280 T^{3} + 40582 T^{4} - 3280 p T^{5} + 332 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 2 T - 48 T^{2} - 224 T^{3} - 10381 T^{4} - 26592 T^{5} + 310252 T^{6} + 3241598 T^{7} + 53462304 T^{8} + 3241598 p T^{9} + 310252 p^{2} T^{10} - 26592 p^{3} T^{11} - 10381 p^{4} T^{12} - 224 p^{5} T^{13} - 48 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.62626484103366144189219218808, −4.56517157378872410049369644520, −4.45914585884115818392003196356, −4.10523337064175282566234090651, −4.01164421594048909148971335445, −3.96983267550546519952688074784, −3.96843107493488924941234525145, −3.69880402016730502571854747988, −3.60899904683549633348664958428, −3.23481107084998574504496773767, −3.10484389604246262839017378428, −3.02030198324333678972406059860, −2.85640863516264206578258713863, −2.32224484461190589782684471885, −2.27834248342654387553550315406, −2.12841428475819288349096045225, −2.09535438998476674957907092221, −1.83200772442889406083912047009, −1.45360927553063656583895475401, −1.42923992882220240047198440951, −1.34668648562098888133853189551, −1.24821117042180304885928083041, −0.77612964108909334366187999742, −0.68924996627948437687227536482, −0.31078450783809097514467536341,
0.31078450783809097514467536341, 0.68924996627948437687227536482, 0.77612964108909334366187999742, 1.24821117042180304885928083041, 1.34668648562098888133853189551, 1.42923992882220240047198440951, 1.45360927553063656583895475401, 1.83200772442889406083912047009, 2.09535438998476674957907092221, 2.12841428475819288349096045225, 2.27834248342654387553550315406, 2.32224484461190589782684471885, 2.85640863516264206578258713863, 3.02030198324333678972406059860, 3.10484389604246262839017378428, 3.23481107084998574504496773767, 3.60899904683549633348664958428, 3.69880402016730502571854747988, 3.96843107493488924941234525145, 3.96983267550546519952688074784, 4.01164421594048909148971335445, 4.10523337064175282566234090651, 4.45914585884115818392003196356, 4.56517157378872410049369644520, 4.62626484103366144189219218808
Plot not available for L-functions of degree greater than 10.
