| L(s) = 1 | − 4·2-s + 6·4-s − 8·9-s − 16·11-s − 15·16-s + 32·18-s + 64·22-s + 4·23-s + 4·25-s − 20·29-s + 24·32-s − 48·36-s − 16·37-s − 20·43-s − 96·44-s − 16·46-s − 16·50-s + 8·53-s + 80·58-s − 6·64-s + 12·67-s − 48·71-s + 64·74-s − 8·79-s + 30·81-s + 80·86-s + 24·92-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s − 8/3·9-s − 4.82·11-s − 3.75·16-s + 7.54·18-s + 13.6·22-s + 0.834·23-s + 4/5·25-s − 3.71·29-s + 4.24·32-s − 8·36-s − 2.63·37-s − 3.04·43-s − 14.4·44-s − 2.35·46-s − 2.26·50-s + 1.09·53-s + 10.5·58-s − 3/4·64-s + 1.46·67-s − 5.69·71-s + 7.43·74-s − 0.900·79-s + 10/3·81-s + 8.62·86-s + 2.50·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.005029753416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005029753416\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{2} - 23 T^{4} + 44 T^{6} + 496 T^{8} + 44 p^{2} T^{10} - 23 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 8 T^{2} - 105 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 36 T^{2} + 671 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 T + 17 T^{2} + 118 T^{3} - 452 T^{4} + 118 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 10 T + 32 T^{2} + 100 T^{3} + 883 T^{4} + 100 p T^{5} + 32 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T^{2} + 63 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 50 T^{2} + 819 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 4 T^{2} + 100 T^{3} + 3067 T^{4} + 100 p T^{5} + 4 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 124 T^{2} + 7354 T^{4} - 446896 T^{6} + 25507219 T^{8} - 446896 p^{2} T^{10} + 7354 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 2 T - 28 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 160 T^{2} + 13198 T^{4} - 870400 T^{6} + 52578643 T^{8} - 870400 p^{2} T^{10} + 13198 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 204 T^{2} + 24145 T^{4} - 2045916 T^{6} + 136536864 T^{8} - 2045916 p^{2} T^{10} + 24145 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 6 T - 92 T^{2} + 36 T^{3} + 9483 T^{4} + 36 p T^{5} - 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 12 T + 163 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 36 T^{2} - 778 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4 T - 131 T^{2} - 44 T^{3} + 14104 T^{4} - 44 p T^{5} - 131 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 256 T^{2} + 36334 T^{4} - 3948544 T^{6} + 353820979 T^{8} - 3948544 p^{2} T^{10} + 36334 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 156 T^{2} + 1594 T^{4} + 612144 T^{6} + 199691859 T^{8} + 612144 p^{2} T^{10} + 1594 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.51091266574658026852749863081, −4.36461293611791741386333111489, −4.22819392347535018427962104358, −3.93058970097253827416468894349, −3.53687099450609946707433716153, −3.50203301308765639054205787130, −3.36751916540407071651609168303, −3.25827116116025203580688435793, −3.22565257202840119191373188860, −3.19674746244636255027984640686, −3.02838704758719297496208954640, −2.86014410267444701403920204236, −2.54065432014319957136099580715, −2.42034476927706105155928745164, −2.33705581447460825180888941396, −1.93167489923818872461381669845, −1.86580650163803274407110774488, −1.85038629750974346873345452327, −1.80140645241199436282320267297, −1.73806838896043113632874819015, −0.937175470315551712895235315786, −0.73942551458320884294839508177, −0.56434663958296403578489086884, −0.36114805893465838179565067041, −0.05022821053127145607746697280,
0.05022821053127145607746697280, 0.36114805893465838179565067041, 0.56434663958296403578489086884, 0.73942551458320884294839508177, 0.937175470315551712895235315786, 1.73806838896043113632874819015, 1.80140645241199436282320267297, 1.85038629750974346873345452327, 1.86580650163803274407110774488, 1.93167489923818872461381669845, 2.33705581447460825180888941396, 2.42034476927706105155928745164, 2.54065432014319957136099580715, 2.86014410267444701403920204236, 3.02838704758719297496208954640, 3.19674746244636255027984640686, 3.22565257202840119191373188860, 3.25827116116025203580688435793, 3.36751916540407071651609168303, 3.50203301308765639054205787130, 3.53687099450609946707433716153, 3.93058970097253827416468894349, 4.22819392347535018427962104358, 4.36461293611791741386333111489, 4.51091266574658026852749863081
Plot not available for L-functions of degree greater than 10.
