| L(s) = 1 | + 2·2-s + 4-s − 4·8-s − 4·9-s + 10·11-s − 6·16-s − 8·18-s + 20·22-s + 4·23-s − 16·25-s + 12·29-s − 4·32-s − 4·36-s + 40·37-s − 8·43-s + 10·44-s + 8·46-s − 32·50-s + 16·53-s + 24·58-s + 6·64-s − 4·67-s + 36·71-s + 16·72-s + 80·74-s + 8·79-s − 10·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.41·8-s − 4/3·9-s + 3.01·11-s − 3/2·16-s − 1.88·18-s + 4.26·22-s + 0.834·23-s − 3.19·25-s + 2.22·29-s − 0.707·32-s − 2/3·36-s + 6.57·37-s − 1.21·43-s + 1.50·44-s + 1.17·46-s − 4.52·50-s + 2.19·53-s + 3.15·58-s + 3/4·64-s − 0.488·67-s + 4.27·71-s + 1.88·72-s + 9.29·74-s + 0.900·79-s − 1.11·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.56710624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.56710624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( ( 1 - T )^{10} \) |
| good | 2 | \( ( 1 - T + T^{2} + T^{3} - p T^{4} + p^{2} T^{5} - p^{2} T^{6} + p^{2} T^{7} + p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 3 | \( 1 + 4 T^{2} + 26 T^{4} + 20 p T^{6} + 67 p T^{8} + 400 T^{10} + 67 p^{3} T^{12} + 20 p^{5} T^{14} + 26 p^{6} T^{16} + 4 p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 + 16 T^{2} + 26 p T^{4} + 936 T^{6} + 6049 T^{8} + 32432 T^{10} + 6049 p^{2} T^{12} + 936 p^{4} T^{14} + 26 p^{7} T^{16} + 16 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( 1 + 58 T^{2} + 1813 T^{4} + 38552 T^{6} + 641042 T^{8} + 8931996 T^{10} + 641042 p^{2} T^{12} + 38552 p^{4} T^{14} + 1813 p^{6} T^{16} + 58 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 + 2 p T^{2} + 701 T^{4} + 9880 T^{6} + 138722 T^{8} + 2623628 T^{10} + 138722 p^{2} T^{12} + 9880 p^{4} T^{14} + 701 p^{6} T^{16} + 2 p^{9} T^{18} + p^{10} T^{20} \) |
| 19 | \( 1 + 46 T^{2} + 1669 T^{4} + 33672 T^{6} + 663394 T^{8} + 10333268 T^{10} + 663394 p^{2} T^{12} + 33672 p^{4} T^{14} + 1669 p^{6} T^{16} + 46 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( ( 1 - 2 T + 76 T^{2} - 58 T^{3} + 2627 T^{4} - 784 T^{5} + 2627 p T^{6} - 58 p^{2} T^{7} + 76 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 - 6 T + 89 T^{2} - 440 T^{3} + 4066 T^{4} - 15204 T^{5} + 4066 p T^{6} - 440 p^{2} T^{7} + 89 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( 1 + 156 T^{2} + 13522 T^{4} + 807092 T^{6} + 36209969 T^{8} + 1264576016 T^{10} + 36209969 p^{2} T^{12} + 807092 p^{4} T^{14} + 13522 p^{6} T^{16} + 156 p^{8} T^{18} + p^{10} T^{20} \) |
| 37 | \( ( 1 - 20 T + 298 T^{2} - 2994 T^{3} + 25477 T^{4} - 167268 T^{5} + 25477 p T^{6} - 2994 p^{2} T^{7} + 298 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 41 | \( 1 + 242 T^{2} + 26221 T^{4} + 1708760 T^{6} + 79854498 T^{8} + 3281591788 T^{10} + 79854498 p^{2} T^{12} + 1708760 p^{4} T^{14} + 26221 p^{6} T^{16} + 242 p^{8} T^{18} + p^{10} T^{20} \) |
| 43 | \( ( 1 + 4 T + 159 T^{2} + 640 T^{3} + 11522 T^{4} + 40504 T^{5} + 11522 p T^{6} + 640 p^{2} T^{7} + 159 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 47 | \( 1 + 100 T^{2} + 12773 T^{4} + 860144 T^{6} + 60083402 T^{8} + 2824202520 T^{10} + 60083402 p^{2} T^{12} + 860144 p^{4} T^{14} + 12773 p^{6} T^{16} + 100 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( ( 1 - 8 T + 137 T^{2} - 832 T^{3} + 9530 T^{4} - 54512 T^{5} + 9530 p T^{6} - 832 p^{2} T^{7} + 137 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 59 | \( 1 + 356 T^{2} + 66666 T^{4} + 8245292 T^{6} + 740132921 T^{8} + 50009679536 T^{10} + 740132921 p^{2} T^{12} + 8245292 p^{4} T^{14} + 66666 p^{6} T^{16} + 356 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( 1 + 466 T^{2} + 104101 T^{4} + 14630744 T^{6} + 1431084002 T^{8} + 101749491180 T^{10} + 1431084002 p^{2} T^{12} + 14630744 p^{4} T^{14} + 104101 p^{6} T^{16} + 466 p^{8} T^{18} + p^{10} T^{20} \) |
| 67 | \( ( 1 + 2 T + 232 T^{2} + 266 T^{3} + 25767 T^{4} + 22336 T^{5} + 25767 p T^{6} + 266 p^{2} T^{7} + 232 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 71 | \( ( 1 - 18 T + 380 T^{2} - 4594 T^{3} + 55299 T^{4} - 470784 T^{5} + 55299 p T^{6} - 4594 p^{2} T^{7} + 380 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 + 330 T^{2} + 53005 T^{4} + 5774264 T^{6} + 507265858 T^{8} + 39095718780 T^{10} + 507265858 p^{2} T^{12} + 5774264 p^{4} T^{14} + 53005 p^{6} T^{16} + 330 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 - 4 T + 279 T^{2} - 1296 T^{3} + 35846 T^{4} - 155544 T^{5} + 35846 p T^{6} - 1296 p^{2} T^{7} + 279 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 + 542 T^{2} + 145909 T^{4} + 25508264 T^{6} + 3212013298 T^{8} + 304929585716 T^{10} + 3212013298 p^{2} T^{12} + 25508264 p^{4} T^{14} + 145909 p^{6} T^{16} + 542 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( 1 + 856 T^{2} + 332602 T^{4} + 77249184 T^{6} + 11876545129 T^{8} + 1262945061008 T^{10} + 11876545129 p^{2} T^{12} + 77249184 p^{4} T^{14} + 332602 p^{6} T^{16} + 856 p^{8} T^{18} + p^{10} T^{20} \) |
| 97 | \( 1 + 744 T^{2} + 260570 T^{4} + 57066864 T^{6} + 8728561161 T^{8} + 980530116880 T^{10} + 8728561161 p^{2} T^{12} + 57066864 p^{4} T^{14} + 260570 p^{6} T^{16} + 744 p^{8} T^{18} + p^{10} T^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.08779771422972611900721023135, −4.03463411177253111473077227838, −3.93302146905185047183585991821, −3.85828791254360964695939080971, −3.41089632346875769139626877105, −3.39527825713004395826329299551, −3.37486869381538944442350586087, −3.35349437189667357625706749500, −3.07584986966519302430491849397, −3.06257061912243831615812289708, −2.95366520781652709251312131442, −2.92059387744415445777463427238, −2.36545606481178679314379470923, −2.36083669060303817333346488297, −2.25386513371403658198030725408, −2.24655082177986904060532131087, −2.23646708287444702920912193916, −1.87096501944512604345109591220, −1.77516287553698706359422148543, −1.39207966057595040646058416832, −1.08292731508802072619681566498, −1.01383440042592199891709032449, −0.934402175737401977273665877553, −0.59817937591525301950136419527, −0.47087446010491116405281159648,
0.47087446010491116405281159648, 0.59817937591525301950136419527, 0.934402175737401977273665877553, 1.01383440042592199891709032449, 1.08292731508802072619681566498, 1.39207966057595040646058416832, 1.77516287553698706359422148543, 1.87096501944512604345109591220, 2.23646708287444702920912193916, 2.24655082177986904060532131087, 2.25386513371403658198030725408, 2.36083669060303817333346488297, 2.36545606481178679314379470923, 2.92059387744415445777463427238, 2.95366520781652709251312131442, 3.06257061912243831615812289708, 3.07584986966519302430491849397, 3.35349437189667357625706749500, 3.37486869381538944442350586087, 3.39527825713004395826329299551, 3.41089632346875769139626877105, 3.85828791254360964695939080971, 3.93302146905185047183585991821, 4.03463411177253111473077227838, 4.08779771422972611900721023135
Plot not available for L-functions of degree greater than 10.
