L(s) = 1 | − 16·13-s + 20·25-s + 64·29-s + 112·37-s + 16·41-s + 168·49-s + 352·53-s + 176·61-s − 240·73-s − 80·89-s + 432·97-s − 224·101-s − 368·109-s + 288·113-s + 552·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 216·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 1.23·13-s + 4/5·25-s + 2.20·29-s + 3.02·37-s + 0.390·41-s + 24/7·49-s + 6.64·53-s + 2.88·61-s − 3.28·73-s − 0.898·89-s + 4.45·97-s − 2.21·101-s − 3.37·109-s + 2.54·113-s + 4.56·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.27·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(17.59357849\) |
\(L(\frac12)\) |
\(\approx\) |
\(17.59357849\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
good | 7 | \( 1 - 24 p T^{2} + 13404 T^{4} - 690840 T^{6} + 31402310 T^{8} - 690840 p^{4} T^{10} + 13404 p^{8} T^{12} - 24 p^{13} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 276 T^{2} + 48006 T^{4} - 276 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T + 204 T^{2} - 1736 T^{3} - 634 T^{4} - 1736 p^{2} T^{5} + 204 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 732 T^{2} - 3840 T^{3} + 247238 T^{4} - 3840 p^{2} T^{5} + 732 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1192 T^{2} + 901404 T^{4} - 482591000 T^{6} + 198221377670 T^{8} - 482591000 p^{4} T^{10} + 901404 p^{8} T^{12} - 1192 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 616 T^{2} + 755676 T^{4} - 531969752 T^{6} + 267063306566 T^{8} - 531969752 p^{4} T^{10} + 755676 p^{8} T^{12} - 616 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 32 T + 1212 T^{2} - 46048 T^{3} + 1958438 T^{4} - 46048 p^{2} T^{5} + 1212 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 2280 T^{2} + 2108508 T^{4} - 623021400 T^{6} - 295925282362 T^{8} - 623021400 p^{4} T^{10} + 2108508 p^{8} T^{12} - 2280 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 56 T + 3948 T^{2} - 174856 T^{3} + 6816518 T^{4} - 174856 p^{2} T^{5} + 3948 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 8 T + 4924 T^{2} - 33080 T^{3} + 10990150 T^{4} - 33080 p^{2} T^{5} + 4924 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 3976 T^{2} + 15992796 T^{4} - 34977997112 T^{6} + 80511749221766 T^{8} - 34977997112 p^{4} T^{10} + 15992796 p^{8} T^{12} - 3976 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 9640 T^{2} + 45901788 T^{4} - 144992767640 T^{6} + 353863561499078 T^{8} - 144992767640 p^{4} T^{10} + 45901788 p^{8} T^{12} - 9640 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 176 T + 396 p T^{2} - 1644496 T^{3} + 101651558 T^{4} - 1644496 p^{2} T^{5} + 396 p^{5} T^{6} - 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 22952 T^{2} + 243302044 T^{4} - 1557392711960 T^{6} + 6588978912358150 T^{8} - 1557392711960 p^{4} T^{10} + 243302044 p^{8} T^{12} - 22952 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 88 T + 12348 T^{2} - 708776 T^{3} + 62059430 T^{4} - 708776 p^{2} T^{5} + 12348 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 19848 T^{2} + 189118044 T^{4} - 1228457910840 T^{6} + 6186195617725190 T^{8} - 1228457910840 p^{4} T^{10} + 189118044 p^{8} T^{12} - 19848 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 26376 T^{2} + 353772444 T^{4} - 3049343603256 T^{6} + 18245696307579590 T^{8} - 3049343603256 p^{4} T^{10} + 353772444 p^{8} T^{12} - 26376 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 120 T + 19740 T^{2} + 1592520 T^{3} + 158554502 T^{4} + 1592520 p^{2} T^{5} + 19740 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 8040 T^{2} + 72964188 T^{4} - 471847486680 T^{6} + 4278949597529798 T^{8} - 471847486680 p^{4} T^{10} + 72964188 p^{8} T^{12} - 8040 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 18184 T^{2} + 266916444 T^{4} - 2437903506104 T^{6} + 19925145362261510 T^{8} - 2437903506104 p^{4} T^{10} + 266916444 p^{8} T^{12} - 18184 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 40 T + 11100 T^{2} + 192920 T^{3} + 121014662 T^{4} + 192920 p^{2} T^{5} + 11100 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 216 T + 43516 T^{2} - 4942440 T^{3} + 582543750 T^{4} - 4942440 p^{2} T^{5} + 43516 p^{4} T^{6} - 216 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.47581971954528249109909999818, −3.29626983779865809073330007108, −3.23117812675103618019252359158, −3.11316126690055487050242996615, −2.79884041139181357304821867183, −2.69547348008305176985470006788, −2.60655305613767744654625907831, −2.55274790591855405682823675014, −2.48135545068747702309996143610, −2.46619966865577479810685734524, −2.42541325629103031811563914766, −2.19410306159402027282572790299, −1.99810081047598688961593552739, −1.89090875468594688123582562423, −1.71422819424341758564228482540, −1.49344973670192789331363840640, −1.37063959875044915631530590270, −1.12492725636531046397339485223, −0.925805875576685172252377035531, −0.876763668645578937828161946355, −0.812638282382978495079601897709, −0.68265048082619791175785036325, −0.62325216423341969149576869392, −0.37526258179975177899230333491, −0.16037241564525177994469121925,
0.16037241564525177994469121925, 0.37526258179975177899230333491, 0.62325216423341969149576869392, 0.68265048082619791175785036325, 0.812638282382978495079601897709, 0.876763668645578937828161946355, 0.925805875576685172252377035531, 1.12492725636531046397339485223, 1.37063959875044915631530590270, 1.49344973670192789331363840640, 1.71422819424341758564228482540, 1.89090875468594688123582562423, 1.99810081047598688961593552739, 2.19410306159402027282572790299, 2.42541325629103031811563914766, 2.46619966865577479810685734524, 2.48135545068747702309996143610, 2.55274790591855405682823675014, 2.60655305613767744654625907831, 2.69547348008305176985470006788, 2.79884041139181357304821867183, 3.11316126690055487050242996615, 3.23117812675103618019252359158, 3.29626983779865809073330007108, 3.47581971954528249109909999818
Plot not available for L-functions of degree greater than 10.