| L(s) = 1 | + 10·4-s − 8·9-s + 16·11-s + 43·16-s + 24·23-s + 28·25-s − 80·36-s + 8·37-s + 160·44-s + 24·53-s + 110·64-s − 32·67-s − 48·71-s + 36·81-s + 240·92-s − 128·99-s + 280·100-s + 64·113-s + 136·121-s + 127-s + 131-s + 137-s + 139-s − 344·144-s + 80·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 5·4-s − 8/3·9-s + 4.82·11-s + 43/4·16-s + 5.00·23-s + 28/5·25-s − 13.3·36-s + 1.31·37-s + 24.1·44-s + 3.29·53-s + 55/4·64-s − 3.90·67-s − 5.69·71-s + 4·81-s + 25.0·92-s − 12.8·99-s + 28·100-s + 6.02·113-s + 12.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 28.6·144-s + 6.57·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(39.86351155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(39.86351155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{8} \) |
| 7 | \( 1 - 4 T^{4} - 314 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| good | 2 | \( ( 1 - 5 T^{2} + p^{4} T^{4} - 45 T^{6} + 101 T^{8} - 45 p^{2} T^{10} + p^{8} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( ( 1 - 14 T^{2} + 121 T^{4} - 774 T^{6} + 4196 T^{8} - 774 p^{2} T^{10} + 121 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 70 T^{2} + 2381 T^{4} + 51950 T^{6} + 796636 T^{8} + 51950 p^{2} T^{10} + 2381 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 20 T^{2} + 1076 T^{4} + 16300 T^{6} + 457366 T^{8} + 16300 p^{2} T^{10} + 1076 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 102 T^{2} + 5093 T^{4} + 161814 T^{6} + 3616780 T^{8} + 161814 p^{2} T^{10} + 5093 p^{4} T^{12} + 102 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 6 T + 48 T^{2} - 270 T^{3} + 1086 T^{4} - 270 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 158 T^{2} + 12293 T^{4} - 609286 T^{6} + 20954380 T^{8} - 609286 p^{2} T^{10} + 12293 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 108 T^{2} + 7028 T^{4} - 324436 T^{6} + 11297430 T^{8} - 324436 p^{2} T^{10} + 7028 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 2 T + 97 T^{2} - 250 T^{3} + 4636 T^{4} - 250 p T^{5} + 97 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 192 T^{2} + 16508 T^{4} + 888384 T^{6} + 38165830 T^{8} + 888384 p^{2} T^{10} + 16508 p^{4} T^{12} + 192 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 140 T^{2} + 13796 T^{4} - 867860 T^{6} + 43988886 T^{8} - 867860 p^{2} T^{10} + 13796 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 230 T^{2} + 28521 T^{4} - 2266390 T^{6} + 126631796 T^{8} - 2266390 p^{2} T^{10} + 28521 p^{4} T^{12} - 230 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 6 T + 188 T^{2} - 850 T^{3} + 14486 T^{4} - 850 p T^{5} + 188 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 278 T^{2} + 40073 T^{4} - 3829126 T^{6} + 263019940 T^{8} - 3829126 p^{2} T^{10} + 40073 p^{4} T^{12} - 278 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 312 T^{2} + 48988 T^{4} + 5009224 T^{6} + 360566630 T^{8} + 5009224 p^{2} T^{10} + 48988 p^{4} T^{12} + 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 8 T + 207 T^{2} + 1600 T^{3} + 18936 T^{4} + 1600 p T^{5} + 207 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 12 T + 248 T^{2} + 1804 T^{3} + 23150 T^{4} + 1804 p T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 10 T^{2} + 18381 T^{4} - 178370 T^{6} + 140146796 T^{8} - 178370 p^{2} T^{10} + 18381 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 168 T^{2} + 16988 T^{4} - 1063576 T^{6} + 82499910 T^{8} - 1063576 p^{2} T^{10} + 16988 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 180 T^{2} + 21956 T^{4} + 2097260 T^{6} + 156627606 T^{8} + 2097260 p^{2} T^{10} + 21956 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 528 T^{2} + 133148 T^{4} - 20911856 T^{6} + 2232438150 T^{8} - 20911856 p^{2} T^{10} + 133148 p^{4} T^{12} - 528 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 60 T^{2} + 15076 T^{4} - 1727620 T^{6} + 110798006 T^{8} - 1727620 p^{2} T^{10} + 15076 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.40994656879570467308101522616, −3.35683391053267246093987693102, −3.25223240147160240677545032411, −3.11523139324577675015833526664, −3.01990878114180841235744627087, −2.99647186779452091291857601714, −2.95285182693675577226824828217, −2.91373970782534047930272976854, −2.83851538455157053922643525821, −2.76308393440092093586973449869, −2.57917790247032179071831903350, −2.33042899740542984839461411731, −2.27526022097897077730158167410, −2.25403187592937118343221357812, −2.19196838071944143170220499186, −2.14134406153041347560310488633, −1.95754150913041106278678099299, −1.69582882614104178319353524381, −1.59573851592133131692039330499, −1.30557263337476017816154954390, −1.23414832611518512301911100706, −1.18362106346580001229351187779, −1.02087509614568616118788150341, −1.01674134541080024853234932960, −0.819702507730140086831793935414,
0.819702507730140086831793935414, 1.01674134541080024853234932960, 1.02087509614568616118788150341, 1.18362106346580001229351187779, 1.23414832611518512301911100706, 1.30557263337476017816154954390, 1.59573851592133131692039330499, 1.69582882614104178319353524381, 1.95754150913041106278678099299, 2.14134406153041347560310488633, 2.19196838071944143170220499186, 2.25403187592937118343221357812, 2.27526022097897077730158167410, 2.33042899740542984839461411731, 2.57917790247032179071831903350, 2.76308393440092093586973449869, 2.83851538455157053922643525821, 2.91373970782534047930272976854, 2.95285182693675577226824828217, 2.99647186779452091291857601714, 3.01990878114180841235744627087, 3.11523139324577675015833526664, 3.25223240147160240677545032411, 3.35683391053267246093987693102, 3.40994656879570467308101522616
Plot not available for L-functions of degree greater than 10.
