| L(s) = 1 | + 2·7-s + 2·13-s + 16-s + 28·19-s − 14·25-s − 18·31-s + 20·37-s + 2·43-s − 49·49-s + 40·61-s − 4·64-s + 18·67-s + 32·73-s − 16·79-s + 4·91-s − 18·97-s + 76·103-s + 28·109-s + 2·112-s + 30·121-s + 127-s + 131-s + 56·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.554·13-s + 1/4·16-s + 6.42·19-s − 2.79·25-s − 3.23·31-s + 3.28·37-s + 0.304·43-s − 7·49-s + 5.12·61-s − 1/2·64-s + 2.19·67-s + 3.74·73-s − 1.80·79-s + 0.419·91-s − 1.82·97-s + 7.48·103-s + 2.68·109-s + 0.188·112-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 4.85·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 223^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{30} \cdot 223^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(173.3666475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(173.3666475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( ( 1 + T )^{10} \) |
| good | 2 | \( 1 - T^{4} + p^{2} T^{6} - T^{8} - 51 T^{10} - p^{2} T^{12} + p^{6} T^{14} - p^{6} T^{16} + p^{10} T^{20} \) |
| 5 | \( 1 + 14 T^{2} + 29 p T^{4} + 1196 T^{6} + 7748 T^{8} + 42617 T^{10} + 7748 p^{2} T^{12} + 1196 p^{4} T^{14} + 29 p^{7} T^{16} + 14 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( ( 1 - T + 26 T^{2} - 25 T^{3} + 312 T^{4} - 258 T^{5} + 312 p T^{6} - 25 p^{2} T^{7} + 26 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( 1 - 30 T^{2} + 675 T^{4} - 1011 p T^{6} + 152595 T^{8} - 1786968 T^{10} + 152595 p^{2} T^{12} - 1011 p^{5} T^{14} + 675 p^{6} T^{16} - 30 p^{8} T^{18} + p^{10} T^{20} \) |
| 13 | \( ( 1 - T + 48 T^{2} - 57 T^{3} + 1044 T^{4} - 1136 T^{5} + 1044 p T^{6} - 57 p^{2} T^{7} + 48 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 17 | \( 1 + 122 T^{2} + 7025 T^{4} + 255575 T^{6} + 6610221 T^{8} + 128513292 T^{10} + 6610221 p^{2} T^{12} + 255575 p^{4} T^{14} + 7025 p^{6} T^{16} + 122 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 - 14 T + 161 T^{2} - 1173 T^{3} + 7367 T^{4} - 34368 T^{5} + 7367 p T^{6} - 1173 p^{2} T^{7} + 161 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 23 | \( 1 + 44 T^{2} + 1383 T^{4} + 26247 T^{6} + 245679 T^{8} + 4604156 T^{10} + 245679 p^{2} T^{12} + 26247 p^{4} T^{14} + 1383 p^{6} T^{16} + 44 p^{8} T^{18} + p^{10} T^{20} \) |
| 29 | \( 1 + 147 T^{2} + 9854 T^{4} + 443471 T^{6} + 16550956 T^{8} + 527775054 T^{10} + 16550956 p^{2} T^{12} + 443471 p^{4} T^{14} + 9854 p^{6} T^{16} + 147 p^{8} T^{18} + p^{10} T^{20} \) |
| 31 | \( ( 1 + 9 T + 149 T^{2} + 918 T^{3} + 8797 T^{4} + 40018 T^{5} + 8797 p T^{6} + 918 p^{2} T^{7} + 149 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( ( 1 - 10 T + 171 T^{2} - 1052 T^{3} + 10757 T^{4} - 49382 T^{5} + 10757 p T^{6} - 1052 p^{2} T^{7} + 171 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 41 | \( 1 + 264 T^{2} + 34401 T^{4} + 2916006 T^{6} + 178954209 T^{8} + 8351092350 T^{10} + 178954209 p^{2} T^{12} + 2916006 p^{4} T^{14} + 34401 p^{6} T^{16} + 264 p^{8} T^{18} + p^{10} T^{20} \) |
| 43 | \( ( 1 - T + 171 T^{2} - 194 T^{3} + 12905 T^{4} - 12962 T^{5} + 12905 p T^{6} - 194 p^{2} T^{7} + 171 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 47 | \( 1 + 46 T^{2} + 6516 T^{4} + 329978 T^{6} + 23741202 T^{8} + 1003487779 T^{10} + 23741202 p^{2} T^{12} + 329978 p^{4} T^{14} + 6516 p^{6} T^{16} + 46 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( 1 + 312 T^{2} + 39779 T^{4} + 2521675 T^{6} + 74975027 T^{8} + 1537653792 T^{10} + 74975027 p^{2} T^{12} + 2521675 p^{4} T^{14} + 39779 p^{6} T^{16} + 312 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( 1 + 227 T^{2} + 31614 T^{4} + 3161355 T^{6} + 251084076 T^{8} + 16232153162 T^{10} + 251084076 p^{2} T^{12} + 3161355 p^{4} T^{14} + 31614 p^{6} T^{16} + 227 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( ( 1 - 20 T + 424 T^{2} - 4940 T^{3} + 57632 T^{4} - 450998 T^{5} + 57632 p T^{6} - 4940 p^{2} T^{7} + 424 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( ( 1 - 9 T + 245 T^{2} - 1295 T^{3} + 24702 T^{4} - 97099 T^{5} + 24702 p T^{6} - 1295 p^{2} T^{7} + 245 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 71 | \( 1 + 314 T^{2} + 58632 T^{4} + 7694904 T^{6} + 770171082 T^{8} + 61274599142 T^{10} + 770171082 p^{2} T^{12} + 7694904 p^{4} T^{14} + 58632 p^{6} T^{16} + 314 p^{8} T^{18} + p^{10} T^{20} \) |
| 73 | \( ( 1 - 16 T + 270 T^{2} - 2968 T^{3} + 35020 T^{4} - 295021 T^{5} + 35020 p T^{6} - 2968 p^{2} T^{7} + 270 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 79 | \( ( 1 + 8 T + 312 T^{2} + 1996 T^{3} + 42980 T^{4} + 218596 T^{5} + 42980 p T^{6} + 1996 p^{2} T^{7} + 312 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 + 546 T^{2} + 146613 T^{4} + 25649112 T^{6} + 3240918066 T^{8} + 308308003404 T^{10} + 3240918066 p^{2} T^{12} + 25649112 p^{4} T^{14} + 146613 p^{6} T^{16} + 546 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( 1 + 339 T^{2} + 49023 T^{4} + 3456246 T^{6} + 47297907 T^{8} - 8789463804 T^{10} + 47297907 p^{2} T^{12} + 3456246 p^{4} T^{14} + 49023 p^{6} T^{16} + 339 p^{8} T^{18} + p^{10} T^{20} \) |
| 97 | \( ( 1 + 9 T + 403 T^{2} + 2926 T^{3} + 71715 T^{4} + 397464 T^{5} + 71715 p T^{6} + 2926 p^{2} T^{7} + 403 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.77836034833280582619996371436, −2.68276072564212153996064103652, −2.59577864307894176656342943204, −2.48242145270344329300164875264, −2.24983890047930781805402364104, −2.15683917675757258760735335454, −2.10889609853225750536587766169, −2.10555904282164254022306683295, −2.02785769813661891726326953084, −1.99874738656640051572774245418, −1.70565045697186523081128937608, −1.69480053713141616228573972707, −1.56386474667018089881371745883, −1.47473303540007489302292309869, −1.45414728869142396971351846756, −1.41439707889615186811838569120, −1.29940546784884169423044320203, −0.929280277058485814063381438996, −0.78701295711334507448968969256, −0.67612467973747849801140426722, −0.62990074223552554583958599784, −0.59130827969484103403420518458, −0.56814414087343893927844756907, −0.48102259995657881849468723157, −0.36920730723501297867849695509,
0.36920730723501297867849695509, 0.48102259995657881849468723157, 0.56814414087343893927844756907, 0.59130827969484103403420518458, 0.62990074223552554583958599784, 0.67612467973747849801140426722, 0.78701295711334507448968969256, 0.929280277058485814063381438996, 1.29940546784884169423044320203, 1.41439707889615186811838569120, 1.45414728869142396971351846756, 1.47473303540007489302292309869, 1.56386474667018089881371745883, 1.69480053713141616228573972707, 1.70565045697186523081128937608, 1.99874738656640051572774245418, 2.02785769813661891726326953084, 2.10555904282164254022306683295, 2.10889609853225750536587766169, 2.15683917675757258760735335454, 2.24983890047930781805402364104, 2.48242145270344329300164875264, 2.59577864307894176656342943204, 2.68276072564212153996064103652, 2.77836034833280582619996371436
Plot not available for L-functions of degree greater than 10.
