| L(s) = 1 | + 8·2-s + 4·3-s + 36·4-s + 32·6-s + 120·8-s + 6·9-s + 144·12-s + 330·16-s − 16·17-s + 48·18-s + 480·24-s + 8·27-s + 8·29-s + 24·31-s + 792·32-s − 128·34-s + 216·36-s + 24·37-s + 1.32e3·48-s + 28·49-s − 64·51-s + 64·54-s + 64·58-s + 192·62-s + 1.71e3·64-s + 8·67-s − 576·68-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 2.30·3-s + 18·4-s + 13.0·6-s + 42.4·8-s + 2·9-s + 41.5·12-s + 82.5·16-s − 3.88·17-s + 11.3·18-s + 97.9·24-s + 1.53·27-s + 1.48·29-s + 4.31·31-s + 140.·32-s − 21.9·34-s + 36·36-s + 3.94·37-s + 190.·48-s + 4·49-s − 8.96·51-s + 8.70·54-s + 8.40·58-s + 24.3·62-s + 214.5·64-s + 0.977·67-s − 69.8·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1184.827420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1184.827420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 - 4 T + 10 T^{2} - 8 p T^{3} + 16 p T^{4} - 8 p^{2} T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 14 T^{4} - 192 T^{6} + 51 T^{8} - 192 p^{2} T^{10} + 14 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 p T^{2} + 452 T^{4} - 5004 T^{6} + 40490 T^{8} - 5004 p^{2} T^{10} + 452 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 36 T^{2} + 418 T^{4} + 2592 T^{6} - 100725 T^{8} + 2592 p^{2} T^{10} + 418 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 8 T + 60 T^{2} + 312 T^{3} + 1363 T^{4} + 312 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 84 T^{2} + 3844 T^{4} - 118404 T^{6} + 2616138 T^{8} - 118404 p^{2} T^{10} + 3844 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 60 T^{2} + 2612 T^{4} - 87180 T^{6} + 2186346 T^{8} - 87180 p^{2} T^{10} + 2612 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 4 T + 78 T^{2} - 264 T^{3} + 2851 T^{4} - 264 p T^{5} + 78 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 154 T^{2} - 1080 T^{3} + 7608 T^{4} - 1080 p T^{5} + 154 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 172 T^{2} - 1224 T^{3} + 9747 T^{4} - 1224 p T^{5} + 172 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 44 T^{2} + 144 T^{3} + 1515 T^{4} + 144 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 232 T^{2} + 24668 T^{4} - 1646616 T^{6} + 80449574 T^{8} - 1646616 p^{2} T^{10} + 24668 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 228 T^{2} + 27716 T^{4} - 2177844 T^{6} + 121057578 T^{8} - 2177844 p^{2} T^{10} + 27716 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 240 T^{2} + 29870 T^{4} - 2491584 T^{6} + 152740371 T^{8} - 2491584 p^{2} T^{10} + 29870 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 336 T^{2} + 53804 T^{4} - 5435952 T^{6} + 380644038 T^{8} - 5435952 p^{2} T^{10} + 53804 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 232 T^{2} + 27620 T^{4} - 2298936 T^{6} + 153904550 T^{8} - 2298936 p^{2} T^{10} + 27620 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 4 T + 170 T^{2} - 648 T^{3} + 15680 T^{4} - 648 p T^{5} + 170 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 408 T^{2} + 79516 T^{4} - 9832104 T^{6} + 848678022 T^{8} - 9832104 p^{2} T^{10} + 79516 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 228 T^{2} + 27316 T^{4} - 3032532 T^{6} + 285218922 T^{8} - 3032532 p^{2} T^{10} + 27316 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 8 T + 342 T^{2} + 1980 T^{3} + 42976 T^{4} + 1980 p T^{5} + 342 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 636 T^{2} + 2050 p T^{4} - 30885600 T^{6} + 3378509787 T^{8} - 30885600 p^{2} T^{10} + 2050 p^{5} T^{12} - 636 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 4 T + 242 T^{2} + 480 T^{3} + 29531 T^{4} + 480 p T^{5} + 242 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.43465613414031300823724367433, −4.41416811811603748501655502615, −4.22585039232628470155845160653, −4.05178219328126506197392289011, −4.03805579530595611895864298546, −3.84323710362140225749534988145, −3.79298823974575726004502533802, −3.72258480515201660415968894720, −3.70287973949309138807954706624, −3.07248076121517531459116250371, −2.98390094236134816784274445469, −2.87057789869377786920446389522, −2.76475249809097062687407184233, −2.75440608143326600597057949980, −2.74913830221045214180243865425, −2.56787731913816329681180127546, −2.54550565593050437633691935332, −2.40171577892978755899423595613, −2.07158923871801570059402762740, −1.87762101975987924897393414743, −1.86878844962873317317251780147, −1.43358601923576908224260071861, −1.17030458405160192610607857378, −0.908520824478621003809393827212, −0.797383060200823120175652271629,
0.797383060200823120175652271629, 0.908520824478621003809393827212, 1.17030458405160192610607857378, 1.43358601923576908224260071861, 1.86878844962873317317251780147, 1.87762101975987924897393414743, 2.07158923871801570059402762740, 2.40171577892978755899423595613, 2.54550565593050437633691935332, 2.56787731913816329681180127546, 2.74913830221045214180243865425, 2.75440608143326600597057949980, 2.76475249809097062687407184233, 2.87057789869377786920446389522, 2.98390094236134816784274445469, 3.07248076121517531459116250371, 3.70287973949309138807954706624, 3.72258480515201660415968894720, 3.79298823974575726004502533802, 3.84323710362140225749534988145, 4.03805579530595611895864298546, 4.05178219328126506197392289011, 4.22585039232628470155845160653, 4.41416811811603748501655502615, 4.43465613414031300823724367433
Plot not available for L-functions of degree greater than 10.
