| L(s) = 1 | + 10·2-s − 3-s + 55·4-s − 10·6-s − 2·7-s + 220·8-s + 2·9-s − 4·11-s − 55·12-s − 20·14-s + 715·16-s + 20·18-s − 6·19-s + 2·21-s − 40·22-s + 8·23-s − 220·24-s + 27·25-s − 5·27-s − 110·28-s + 2.00e3·32-s + 4·33-s + 110·36-s − 60·38-s + 20·42-s − 220·44-s + 80·46-s + ⋯ |
| L(s) = 1 | + 7.07·2-s − 0.577·3-s + 55/2·4-s − 4.08·6-s − 0.755·7-s + 77.7·8-s + 2/3·9-s − 1.20·11-s − 15.8·12-s − 5.34·14-s + 178.·16-s + 4.71·18-s − 1.37·19-s + 0.436·21-s − 8.52·22-s + 1.66·23-s − 44.9·24-s + 27/5·25-s − 0.962·27-s − 20.7·28-s + 353.·32-s + 0.696·33-s + 55/3·36-s − 9.73·38-s + 3.08·42-s − 33.1·44-s + 11.7·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(515.7900996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(515.7900996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 3 | \( 1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 2 p T^{6} + 2 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | \( 1 + 20 T + 71 T^{2} - 896 T^{3} - 6134 T^{4} - 9960 T^{5} - 6134 p T^{6} - 896 p^{2} T^{7} + 71 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 5 | \( 1 - 27 T^{2} + 79 p T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 29068 p^{2} T^{12} - 3938 p^{4} T^{14} + 79 p^{7} T^{16} - 27 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( ( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 179 p T^{6} + 17 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( ( 1 + 2 T + 30 T^{2} + 34 T^{3} + 439 T^{4} + 372 T^{5} + 439 p T^{6} + 34 p^{2} T^{7} + 30 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 4 p T^{2} + 88 p T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 57459 p^{2} T^{12} - 10254 p^{4} T^{14} + 88 p^{7} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 3938173 p^{2} T^{12} - 171278 p^{4} T^{14} + 5426 p^{6} T^{16} - 108 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 2608 p T^{6} + 178 p^{2} T^{7} + 77 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 23 | \( ( 1 - 4 T + 29 T^{2} - 188 T^{3} + 1084 T^{4} - 2688 T^{5} + 1084 p T^{6} - 188 p^{2} T^{7} + 29 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( 1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 30825980 p^{2} T^{12} - 729682 p^{4} T^{14} + 12907 p^{6} T^{16} - 155 p^{8} T^{18} + p^{10} T^{20} \) |
| 31 | \( 1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2683060 p^{2} T^{12} - 68420 p^{4} T^{14} + 2027 p^{6} T^{16} - 66 p^{8} T^{18} + p^{10} T^{20} \) |
| 37 | \( 1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 9281381 p^{2} T^{12} - 563614 p^{4} T^{14} + 14592 p^{6} T^{16} - 188 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( 1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 106670111 p^{2} T^{12} - 1817599 p^{4} T^{14} + 23428 p^{6} T^{16} - 209 p^{8} T^{18} + p^{10} T^{20} \) |
| 43 | \( 1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 29482965 p^{2} T^{12} - 558234 p^{4} T^{14} + 8986 p^{6} T^{16} - 124 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( ( 1 + 149 T^{2} - 36 T^{3} + 11260 T^{4} - 2520 T^{5} + 11260 p T^{6} - 36 p^{2} T^{7} + 149 p^{3} T^{8} + p^{5} T^{10} )^{2} \) |
| 53 | \( 1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 540355392 p^{2} T^{12} - 6738486 p^{4} T^{14} + 60783 p^{6} T^{16} - 355 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( 1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 216253284 p^{2} T^{12} - 2771220 p^{4} T^{14} + 28531 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20} \) |
| 67 | \( 1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 180779250 p^{2} T^{12} - 2099544 p^{4} T^{14} + 24469 p^{6} T^{16} - 226 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( 1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 1288692677 p^{2} T^{12} - 12232858 p^{4} T^{14} + 85792 p^{6} T^{16} - 404 p^{8} T^{18} + p^{10} T^{20} \) |
| 73 | \( 1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 488701726 p^{2} T^{12} - 4874480 p^{4} T^{14} + 40625 p^{6} T^{16} - 234 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 24163 p T^{6} - 299 p^{2} T^{7} + 192 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( ( 1 + 6 T + 260 T^{2} + 1794 T^{3} + 33211 T^{4} + 211464 T^{5} + 33211 p T^{6} + 1794 p^{2} T^{7} + 260 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 89 | \( ( 1 - 8 T + 201 T^{2} - 220 T^{3} + 15682 T^{4} + 31200 T^{5} + 15682 p T^{6} - 220 p^{2} T^{7} + 201 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 7388315838 p^{2} T^{12} - 48709872 p^{4} T^{14} + 227521 p^{6} T^{16} - 682 p^{8} T^{18} + p^{10} T^{20} \) |
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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
−4.48356315150511407749536223809, −4.40600749912849691755124409561, −4.32130223328342972547538240265, −4.14741281119110861241262499869, −3.94384903629818456749368003547, −3.85920465020403360410838660233, −3.58282178966546153115792954030, −3.55156126963239909157157162021, −3.41766909118895374621144446258, −3.25714664459595542053356796462, −3.11604809754873204705706354911, −3.08293911131522132672012767328, −3.03005036825759810604996539132, −3.00259817406281952536928454367, −2.61217880368033382861039027046, −2.59047343988744707823326663436, −2.56215397900690039640870288530, −2.25053679178613362927581319599, −2.21577488532363537192228575653, −1.86838842899287159893391551978, −1.74147727734882790774043167309, −1.36120296904609115429053525071, −1.33229144942458229243096498656, −1.31552356390640228712936215521, −0.67753741348863682419101473221,
0.67753741348863682419101473221, 1.31552356390640228712936215521, 1.33229144942458229243096498656, 1.36120296904609115429053525071, 1.74147727734882790774043167309, 1.86838842899287159893391551978, 2.21577488532363537192228575653, 2.25053679178613362927581319599, 2.56215397900690039640870288530, 2.59047343988744707823326663436, 2.61217880368033382861039027046, 3.00259817406281952536928454367, 3.03005036825759810604996539132, 3.08293911131522132672012767328, 3.11604809754873204705706354911, 3.25714664459595542053356796462, 3.41766909118895374621144446258, 3.55156126963239909157157162021, 3.58282178966546153115792954030, 3.85920465020403360410838660233, 3.94384903629818456749368003547, 4.14741281119110861241262499869, 4.32130223328342972547538240265, 4.40600749912849691755124409561, 4.48356315150511407749536223809
Plot not available for L-functions of degree greater than 10.
