L(s) = 1 | − 9-s + 20·11-s + 12·29-s + 44·31-s − 12·41-s + 48·49-s − 12·59-s − 14·61-s + 18·71-s + 64·79-s + 81-s + 4·89-s − 20·99-s + 22·101-s + 46·109-s + 118·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 75·169-s + 173-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 6.03·11-s + 2.22·29-s + 7.90·31-s − 1.87·41-s + 48/7·49-s − 1.56·59-s − 1.79·61-s + 2.13·71-s + 7.20·79-s + 1/9·81-s + 0.423·89-s − 2.01·99-s + 2.18·101-s + 4.40·109-s + 10.7·121-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 − 5.76·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(103.1039284\) |
\(L(\frac12)\) |
\(\approx\) |
\(103.1039284\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - 18 T^{2} - 7 T^{3} - 18 p T^{4} + p^{3} T^{6} )^{2} \) |
good | 3 | \( 1 + T^{2} - 65 T^{6} - 41 T^{8} + 32 p T^{10} + 91 p^{3} T^{12} + 32 p^{3} T^{14} - 41 p^{4} T^{16} - 65 p^{6} T^{18} + p^{10} T^{22} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 24 T^{2} + 300 T^{4} - 2509 T^{6} + 300 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 5 T + 3 p T^{2} - 101 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 13 | \( ( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 17 | \( 1 + 21 T^{2} - 168 T^{4} - 10877 T^{6} - 53613 T^{8} + 1568196 T^{10} + 44277657 T^{12} + 1568196 p^{2} T^{14} - 53613 p^{4} T^{16} - 10877 p^{6} T^{18} - 168 p^{8} T^{20} + 21 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( 1 + 20 T^{2} + 540 T^{4} + 7782 T^{6} + 199480 T^{8} + 9299960 T^{10} + 201761659 T^{12} + 9299960 p^{2} T^{14} + 199480 p^{4} T^{16} + 7782 p^{6} T^{18} + 540 p^{8} T^{20} + 20 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( ( 1 - 6 T + 30 T^{2} - 366 T^{3} + 852 T^{4} - 1968 T^{5} + 44683 T^{6} - 1968 p T^{7} + 852 p^{2} T^{8} - 366 p^{3} T^{9} + 30 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 11 T + 107 T^{2} - 611 T^{3} + 107 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 37 | \( ( 1 - 204 T^{2} + 17940 T^{4} - 870289 T^{6} + 17940 p^{2} T^{8} - 204 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 + 6 T - 24 T^{2} - 354 T^{3} - 1002 T^{4} + 1986 T^{5} + 57175 T^{6} + 1986 p T^{7} - 1002 p^{2} T^{8} - 354 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( 1 + 109 T^{2} + 6980 T^{4} + 201423 T^{6} - 3178985 T^{8} - 787363480 T^{10} - 44702030183 T^{12} - 787363480 p^{2} T^{14} - 3178985 p^{4} T^{16} + 201423 p^{6} T^{18} + 6980 p^{8} T^{20} + 109 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 84 T^{2} - 1572 T^{4} - 44042 T^{6} + 23093784 T^{8} + 524516184 T^{10} - 29554771125 T^{12} + 524516184 p^{2} T^{14} + 23093784 p^{4} T^{16} - 44042 p^{6} T^{18} - 1572 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( 1 + 209 T^{2} + 23520 T^{4} + 1692495 T^{6} + 86373079 T^{8} + 3391323344 T^{10} + 146956222537 T^{12} + 3391323344 p^{2} T^{14} + 86373079 p^{4} T^{16} + 1692495 p^{6} T^{18} + 23520 p^{8} T^{20} + 209 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( ( 1 + 6 T - 60 T^{2} + 186 T^{3} + 2382 T^{4} - 28362 T^{5} - 220277 T^{6} - 28362 p T^{7} + 2382 p^{2} T^{8} + 186 p^{3} T^{9} - 60 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 7 T - 117 T^{2} - 420 T^{3} + 11893 T^{4} + 15589 T^{5} - 783602 T^{6} + 15589 p T^{7} + 11893 p^{2} T^{8} - 420 p^{3} T^{9} - 117 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 202 T^{2} + 19949 T^{4} + 1104078 T^{6} + 27341098 T^{8} - 1335923758 T^{10} - 177836720315 T^{12} - 1335923758 p^{2} T^{14} + 27341098 p^{4} T^{16} + 1104078 p^{6} T^{18} + 19949 p^{8} T^{20} + 202 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 - 9 T - 114 T^{2} + 747 T^{3} + 12921 T^{4} - 40104 T^{5} - 827273 T^{6} - 40104 p T^{7} + 12921 p^{2} T^{8} + 747 p^{3} T^{9} - 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 252 T^{2} + 34020 T^{4} + 2637454 T^{6} + 108329256 T^{8} - 1381000824 T^{10} - 391218872061 T^{12} - 1381000824 p^{2} T^{14} + 108329256 p^{4} T^{16} + 2637454 p^{6} T^{18} + 34020 p^{8} T^{20} + 252 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 - 32 T + 479 T^{2} - 5616 T^{3} + 64786 T^{4} - 666232 T^{5} + 6082063 T^{6} - 666232 p T^{7} + 64786 p^{2} T^{8} - 5616 p^{3} T^{9} + 479 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 125 T^{2} + 12925 T^{4} - 835849 T^{6} + 12925 p^{2} T^{8} - 125 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 2 T - 131 T^{2} + 298 T^{3} + 5642 T^{4} - 6122 T^{5} - 163115 T^{6} - 6122 p T^{7} + 5642 p^{2} T^{8} + 298 p^{3} T^{9} - 131 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 + 273 T^{2} + 37740 T^{4} + 2579251 T^{6} - 11573109 T^{8} - 27528917436 T^{10} - 3738113668551 T^{12} - 27528917436 p^{2} T^{14} - 11573109 p^{4} T^{16} + 2579251 p^{6} T^{18} + 37740 p^{8} T^{20} + 273 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.83393314859157944471322213517, −2.71654328277695704338005612130, −2.67852538842759472828819459813, −2.60780804010186684673832290669, −2.58788960151925874124740143674, −2.42336055382173869414259779449, −2.33696404648450825585670993453, −2.13458591754623908246234831743, −2.11719088183230713481288392009, −2.03091643410842390662532621536, −1.86272089282453797816172552924, −1.81858888988176071684316409698, −1.68636311984860860137736944034, −1.58689258098948048546239893077, −1.44471952253781392174823723220, −1.40428581528097337730112197510, −1.04956958593423298981825162783, −0.983756175107212814491582511130, −0.935637067458283935612006885027, −0.931060673336458747354398034377, −0.855131959933540068718877910492, −0.825335686299070068595831762352, −0.75848869116508844107064401442, −0.53778917212391410770072805186, −0.23534179522943985335501346592,
0.23534179522943985335501346592, 0.53778917212391410770072805186, 0.75848869116508844107064401442, 0.825335686299070068595831762352, 0.855131959933540068718877910492, 0.931060673336458747354398034377, 0.935637067458283935612006885027, 0.983756175107212814491582511130, 1.04956958593423298981825162783, 1.40428581528097337730112197510, 1.44471952253781392174823723220, 1.58689258098948048546239893077, 1.68636311984860860137736944034, 1.81858888988176071684316409698, 1.86272089282453797816172552924, 2.03091643410842390662532621536, 2.11719088183230713481288392009, 2.13458591754623908246234831743, 2.33696404648450825585670993453, 2.42336055382173869414259779449, 2.58788960151925874124740143674, 2.60780804010186684673832290669, 2.67852538842759472828819459813, 2.71654328277695704338005612130, 2.83393314859157944471322213517
Plot not available for L-functions of degree greater than 10.