L(s) = 1 | + 3·5-s + 3·7-s + 8-s − 12·11-s − 21·13-s − 3·17-s + 6·19-s + 15·23-s + 9·25-s − 15·29-s + 3·31-s + 9·35-s + 6·37-s + 3·40-s + 9·41-s − 9·43-s + 21·47-s + 15·49-s − 30·53-s − 36·55-s + 3·56-s − 27·59-s − 9·61-s − 63·65-s − 15·67-s − 9·71-s + 12·73-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 1.13·7-s + 0.353·8-s − 3.61·11-s − 5.82·13-s − 0.727·17-s + 1.37·19-s + 3.12·23-s + 9/5·25-s − 2.78·29-s + 0.538·31-s + 1.52·35-s + 0.986·37-s + 0.474·40-s + 1.40·41-s − 1.37·43-s + 3.06·47-s + 15/7·49-s − 4.12·53-s − 4.85·55-s + 0.400·56-s − 3.51·59-s − 1.15·61-s − 7.81·65-s − 1.83·67-s − 1.06·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.116828778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116828778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{3} + T^{6} \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
good | 5 | \( 1 - 3 T - 9 T^{4} + 3 p T^{5} + 109 T^{6} + 3 p^{2} T^{7} - 9 p^{2} T^{8} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 3 T - 6 T^{2} + 41 T^{3} - 9 T^{4} - 162 T^{5} + 519 T^{6} - 162 p T^{7} - 9 p^{2} T^{8} + 41 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 12 T + 6 p T^{2} + 306 T^{3} + 1446 T^{4} + 5430 T^{5} + 17539 T^{6} + 5430 p T^{7} + 1446 p^{2} T^{8} + 306 p^{3} T^{9} + 6 p^{5} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 7 T + p T^{2} )^{3}( 1 - 89 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 + 3 T - 18 T^{2} - 180 T^{3} - 423 T^{4} + 105 p T^{5} + 1097 p T^{6} + 105 p^{2} T^{7} - 423 p^{2} T^{8} - 180 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 15 T + 108 T^{2} - 522 T^{3} + 1881 T^{4} - 6855 T^{5} + 32347 T^{6} - 6855 p T^{7} + 1881 p^{2} T^{8} - 522 p^{3} T^{9} + 108 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 15 T + 72 T^{2} - 36 T^{3} - 729 T^{4} + 13389 T^{5} + 142381 T^{6} + 13389 p T^{7} - 729 p^{2} T^{8} - 36 p^{3} T^{9} + 72 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 3 T - 78 T^{2} + 77 T^{3} + 4365 T^{4} - 1404 T^{5} - 155145 T^{6} - 1404 p T^{7} + 4365 p^{2} T^{8} + 77 p^{3} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 3 T + 102 T^{2} - 203 T^{3} + 102 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - 9 T + 27 T^{2} - 153 T^{3} - 1728 T^{4} + 21024 T^{5} - 78155 T^{6} + 21024 p T^{7} - 1728 p^{2} T^{8} - 153 p^{3} T^{9} + 27 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 9 T + 45 T^{2} + 209 T^{3} - 972 T^{4} - 25110 T^{5} - 174579 T^{6} - 25110 p T^{7} - 972 p^{2} T^{8} + 209 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 21 T + 261 T^{2} - 2673 T^{3} + 22680 T^{4} - 170022 T^{5} + 1202725 T^{6} - 170022 p T^{7} + 22680 p^{2} T^{8} - 2673 p^{3} T^{9} + 261 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 30 T + 378 T^{2} + 2214 T^{3} - 3276 T^{4} - 201552 T^{5} - 2070449 T^{6} - 201552 p T^{7} - 3276 p^{2} T^{8} + 2214 p^{3} T^{9} + 378 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 27 T + 360 T^{2} + 3312 T^{3} + 30879 T^{4} + 316287 T^{5} + 2798533 T^{6} + 316287 p T^{7} + 30879 p^{2} T^{8} + 3312 p^{3} T^{9} + 360 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 9 T + 72 T^{2} + 506 T^{3} + 108 p T^{4} + 59805 T^{5} + 418827 T^{6} + 59805 p T^{7} + 108 p^{3} T^{8} + 506 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 15 T + 156 T^{2} + 1472 T^{3} + 16461 T^{4} + 143415 T^{5} + 1174659 T^{6} + 143415 p T^{7} + 16461 p^{2} T^{8} + 1472 p^{3} T^{9} + 156 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 9 T + 81 T^{2} + 9 T^{3} + 2916 T^{4} - 32418 T^{5} - 92555 T^{6} - 32418 p T^{7} + 2916 p^{2} T^{8} + 9 p^{3} T^{9} + 81 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 12 T + 246 T^{2} - 2623 T^{3} + 30177 T^{4} - 326385 T^{5} + 2687649 T^{6} - 326385 p T^{7} + 30177 p^{2} T^{8} - 2623 p^{3} T^{9} + 246 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 15 T + 105 T^{2} - 559 T^{3} - 6300 T^{4} + 127350 T^{5} - 1087083 T^{6} + 127350 p T^{7} - 6300 p^{2} T^{8} - 559 p^{3} T^{9} + 105 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 3 T + 30 T^{2} - 2727 T^{3} + 6369 T^{4} - 60420 T^{5} + 2990995 T^{6} - 60420 p T^{7} + 6369 p^{2} T^{8} - 2727 p^{3} T^{9} + 30 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 48 T + 1044 T^{2} - 11907 T^{3} + 43551 T^{4} + 760371 T^{5} - 12833135 T^{6} + 760371 p T^{7} + 43551 p^{2} T^{8} - 11907 p^{3} T^{9} + 1044 p^{4} T^{10} - 48 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 18 T + 99 T^{2} + 551 T^{3} - 5967 T^{4} - 122661 T^{5} + 2662518 T^{6} - 122661 p T^{7} - 5967 p^{2} T^{8} + 551 p^{3} T^{9} + 99 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.33866744826431810762125067640, −5.99268438737913470398895939833, −5.69364992419396184046176598801, −5.49750475854216018732727996825, −5.48664641253853594985843447549, −5.41770898861100149597623126160, −4.92812138863414431171367189360, −4.89967447789675559699157898262, −4.89194418656639326084985937241, −4.88074047955053119635819148619, −4.56950585627728649111121170119, −4.49743158354311285248120205491, −4.24550070093103185499811114792, −3.67539310368650726548170507040, −3.19046423039792206033741577423, −3.00046479096867885549956718289, −2.89203696212028323312257847745, −2.88647675183581863557812176145, −2.51608745465890029380443591786, −2.41170637635693180223929278155, −2.05415999510673551794126906891, −1.87249957561237596505139017143, −1.71625126830683767274903728242, −0.840638065484889849145238027212, −0.33971945234757511603759779554,
0.33971945234757511603759779554, 0.840638065484889849145238027212, 1.71625126830683767274903728242, 1.87249957561237596505139017143, 2.05415999510673551794126906891, 2.41170637635693180223929278155, 2.51608745465890029380443591786, 2.88647675183581863557812176145, 2.89203696212028323312257847745, 3.00046479096867885549956718289, 3.19046423039792206033741577423, 3.67539310368650726548170507040, 4.24550070093103185499811114792, 4.49743158354311285248120205491, 4.56950585627728649111121170119, 4.88074047955053119635819148619, 4.89194418656639326084985937241, 4.89967447789675559699157898262, 4.92812138863414431171367189360, 5.41770898861100149597623126160, 5.48664641253853594985843447549, 5.49750475854216018732727996825, 5.69364992419396184046176598801, 5.99268438737913470398895939833, 6.33866744826431810762125067640
Plot not available for L-functions of degree greater than 10.