| L(s) = 1 | + 11·9-s − 8·11-s − 8·19-s + 22·29-s + 8·31-s − 16·41-s + 37·49-s − 46·59-s − 52·61-s − 4·71-s − 86·79-s + 53·81-s − 30·89-s − 88·99-s − 34·101-s − 4·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 55·169-s + ⋯ |
| L(s) = 1 | + 11/3·9-s − 2.41·11-s − 1.83·19-s + 4.08·29-s + 1.43·31-s − 2.49·41-s + 37/7·49-s − 5.98·59-s − 6.65·61-s − 0.474·71-s − 9.67·79-s + 53/9·81-s − 3.17·89-s − 8.84·99-s − 3.38·101-s − 0.383·109-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.23·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7924979551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7924979551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + T^{2} )^{5} \) |
| good | 3 | \( 1 - 11 T^{2} + 68 T^{4} - 32 p^{2} T^{6} + 332 p T^{8} - 3068 T^{10} + 332 p^{3} T^{12} - 32 p^{6} T^{14} + 68 p^{6} T^{16} - 11 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( 1 - 37 T^{2} + 107 p T^{4} - 10348 T^{6} + 106370 T^{8} - 843806 T^{10} + 106370 p^{2} T^{12} - 10348 p^{4} T^{14} + 107 p^{7} T^{16} - 37 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 + 4 T + 38 T^{2} + 8 p T^{3} + 557 T^{4} + 960 T^{5} + 557 p T^{6} + 8 p^{3} T^{7} + 38 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 55 T^{2} + 1912 T^{4} - 46120 T^{6} + 856048 T^{8} - 12464776 T^{10} + 856048 p^{2} T^{12} - 46120 p^{4} T^{14} + 1912 p^{6} T^{16} - 55 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 89 T^{2} + 3789 T^{4} - 107964 T^{6} + 2407058 T^{8} - 44639446 T^{10} + 2407058 p^{2} T^{12} - 107964 p^{4} T^{14} + 3789 p^{6} T^{16} - 89 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 + 4 T + 42 T^{2} + 40 T^{3} + 317 T^{4} - 1304 T^{5} + 317 p T^{6} + 40 p^{2} T^{7} + 42 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 - 11 T + 134 T^{2} - 970 T^{3} + 238 p T^{4} - 37502 T^{5} + 238 p^{2} T^{6} - 970 p^{2} T^{7} + 134 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( ( 1 - 4 T + 80 T^{2} - 337 T^{3} + 3437 T^{4} - 12806 T^{5} + 3437 p T^{6} - 337 p^{2} T^{7} + 80 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( 1 - 54 T^{2} + 3925 T^{4} - 221416 T^{6} + 9303682 T^{8} - 387581700 T^{10} + 9303682 p^{2} T^{12} - 221416 p^{4} T^{14} + 3925 p^{6} T^{16} - 54 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 + 8 T + 153 T^{2} + 641 T^{3} + 8317 T^{4} + 23597 T^{5} + 8317 p T^{6} + 641 p^{2} T^{7} + 153 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 - 133 T^{2} + 12789 T^{4} - 898572 T^{6} + 51067650 T^{8} - 2426678334 T^{10} + 51067650 p^{2} T^{12} - 898572 p^{4} T^{14} + 12789 p^{6} T^{16} - 133 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( 1 - 132 T^{2} + 9454 T^{4} - 478309 T^{6} + 20414825 T^{8} - 904412158 T^{10} + 20414825 p^{2} T^{12} - 478309 p^{4} T^{14} + 9454 p^{6} T^{16} - 132 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( 1 - 102 T^{2} + 11669 T^{4} - 891432 T^{6} + 59048418 T^{8} - 3461934884 T^{10} + 59048418 p^{2} T^{12} - 891432 p^{4} T^{14} + 11669 p^{6} T^{16} - 102 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( ( 1 + 23 T + 348 T^{2} + 4341 T^{3} + 43633 T^{4} + 357240 T^{5} + 43633 p T^{6} + 4341 p^{2} T^{7} + 348 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 61 | \( ( 1 + 26 T + 541 T^{2} + 7192 T^{3} + 81278 T^{4} + 683676 T^{5} + 81278 p T^{6} + 7192 p^{2} T^{7} + 541 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 541 T^{2} + 138253 T^{4} - 21966540 T^{6} + 2401398778 T^{8} - 188631089358 T^{10} + 2401398778 p^{2} T^{12} - 21966540 p^{4} T^{14} + 138253 p^{6} T^{16} - 541 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 + 2 T + 288 T^{2} + 435 T^{3} + 36805 T^{4} + 41598 T^{5} + 36805 p T^{6} + 435 p^{2} T^{7} + 288 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 + 22 T^{2} + 11651 T^{4} - 606563 T^{6} + 18371459 T^{8} - 7921669483 T^{10} + 18371459 p^{2} T^{12} - 606563 p^{4} T^{14} + 11651 p^{6} T^{16} + 22 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 + 43 T + 1029 T^{2} + 16880 T^{3} + 210588 T^{4} + 2084442 T^{5} + 210588 p T^{6} + 16880 p^{2} T^{7} + 1029 p^{3} T^{8} + 43 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 268 T^{2} + 41366 T^{4} - 4873570 T^{6} + 499213561 T^{8} - 44380545380 T^{10} + 499213561 p^{2} T^{12} - 4873570 p^{4} T^{14} + 41366 p^{6} T^{16} - 268 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 + 15 T + 291 T^{2} + 2152 T^{3} + 25484 T^{4} + 1554 p T^{5} + 25484 p T^{6} + 2152 p^{2} T^{7} + 291 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 450 T^{2} + 105757 T^{4} - 1912 p^{2} T^{6} + 2409127842 T^{8} - 259375226764 T^{10} + 2409127842 p^{2} T^{12} - 1912 p^{6} T^{14} + 105757 p^{6} T^{16} - 450 p^{8} T^{18} + p^{10} T^{20} \) |
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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
−2.79974815355376405602585664428, −2.71208327293603857536999308906, −2.59278825920438505816928972653, −2.58142245131451629952350080611, −2.53193034757797898648045646507, −2.49628019861925778036447031836, −2.39242332520993070101829472065, −2.21692322938103891377284903477, −2.11698395623133677032634983056, −2.09130403481138195476734195028, −1.69769897815204959121875810364, −1.60945118630869967200253306140, −1.50519136670070217635022270491, −1.50202874673285258162631824811, −1.39603915132822248243569456485, −1.39448779421601952574778022814, −1.29521148969180505669391860748, −1.28556164745002018135063204824, −1.19047294872526157411443753623, −1.08359982691845161745317578006, −0.65355185044821512872024189177, −0.41737490862760289317709817446, −0.36218022592829274741426318712, −0.20992876490978242350656714206, −0.083464861566401638230865893010,
0.083464861566401638230865893010, 0.20992876490978242350656714206, 0.36218022592829274741426318712, 0.41737490862760289317709817446, 0.65355185044821512872024189177, 1.08359982691845161745317578006, 1.19047294872526157411443753623, 1.28556164745002018135063204824, 1.29521148969180505669391860748, 1.39448779421601952574778022814, 1.39603915132822248243569456485, 1.50202874673285258162631824811, 1.50519136670070217635022270491, 1.60945118630869967200253306140, 1.69769897815204959121875810364, 2.09130403481138195476734195028, 2.11698395623133677032634983056, 2.21692322938103891377284903477, 2.39242332520993070101829472065, 2.49628019861925778036447031836, 2.53193034757797898648045646507, 2.58142245131451629952350080611, 2.59278825920438505816928972653, 2.71208327293603857536999308906, 2.79974815355376405602585664428
Plot not available for L-functions of degree greater than 10.
