| L(s) = 1 | + 84·9-s − 306·25-s − 480·37-s − 3.96e3·67-s + 876·79-s + 3.02e3·81-s − 2.59e3·109-s + 4.03e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.91e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 28/9·9-s − 2.44·25-s − 2.13·37-s − 7.22·67-s + 1.24·79-s + 4.14·81-s − 2.27·109-s + 3.03·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.32·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.487681964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.487681964\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 28 p T^{2} + 448 p^{2} T^{4} - 4730 p^{3} T^{6} + 448 p^{8} T^{8} - 28 p^{13} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 153 T^{2} + 19386 T^{4} + 3844681 T^{6} + 19386 p^{6} T^{8} + 153 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 11 | \( ( 1 - 2019 T^{2} + 2209074 T^{4} - 312678449 p T^{6} + 2209074 p^{6} T^{8} - 2019 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 13 | \( ( 1 - 1458 T^{2} + 12022119 T^{4} - 10536506876 T^{6} + 12022119 p^{6} T^{8} - 1458 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 17 | \( ( 1 + 9576 T^{2} + 69763680 T^{4} + 456896326822 T^{6} + 69763680 p^{6} T^{8} + 9576 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 19 | \( ( 1 - 25752 T^{2} + 340350888 T^{4} - 2857540917530 T^{6} + 340350888 p^{6} T^{8} - 25752 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 23 | \( ( 1 - 41664 T^{2} + 733970328 T^{4} - 9126533599738 T^{6} + 733970328 p^{6} T^{8} - 41664 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 29 | \( ( 1 - 90957 T^{2} + 3685482627 T^{4} - 100535121841678 T^{6} + 3685482627 p^{6} T^{8} - 90957 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 31 | \( ( 1 - 93489 T^{2} + 4001732286 T^{4} - 123118448290229 T^{6} + 4001732286 p^{6} T^{8} - 93489 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 37 | \( ( 1 + 120 T + 145848 T^{2} + 11670122 T^{3} + 145848 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6} )^{4} \) |
| 41 | \( ( 1 + 284094 T^{2} + 38573492751 T^{4} + 3254645758116292 T^{6} + 38573492751 p^{6} T^{8} + 284094 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 43 | \( ( 1 + 137673 T^{2} + 12209024 T^{3} + 137673 p^{3} T^{4} + p^{9} T^{6} )^{4} \) |
| 47 | \( ( 1 + 90936 T^{2} - 1506355848 T^{4} - 1399368809936306 T^{6} - 1506355848 p^{6} T^{8} + 90936 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 53 | \( ( 1 - 289047 T^{2} + 15753021018 T^{4} + 1536442315252121 T^{6} + 15753021018 p^{6} T^{8} - 289047 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 59 | \( ( 1 + 682605 T^{2} + 263615723154 T^{4} + 66101550285851701 T^{6} + 263615723154 p^{6} T^{8} + 682605 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 61 | \( ( 1 - 937536 T^{2} + 420448820784 T^{4} - 116963888204990810 T^{6} + 420448820784 p^{6} T^{8} - 937536 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 67 | \( ( 1 + 990 T + 874908 T^{2} + 488695102 T^{3} + 874908 p^{3} T^{4} + 990 p^{6} T^{5} + p^{9} T^{6} )^{4} \) |
| 71 | \( ( 1 - 1325730 T^{2} + 947795292591 T^{4} - 415349195473127932 T^{6} + 947795292591 p^{6} T^{8} - 1325730 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 73 | \( ( 1 - 1744752 T^{2} + 1427886700608 T^{4} - 699289244346941570 T^{6} + 1427886700608 p^{6} T^{8} - 1744752 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 79 | \( ( 1 - 219 T + 1277112 T^{2} - 166970531 T^{3} + 1277112 p^{3} T^{4} - 219 p^{6} T^{5} + p^{9} T^{6} )^{4} \) |
| 83 | \( ( 1 + 2213631 T^{2} + 2402637322995 T^{4} + 1655758504150116154 T^{6} + 2402637322995 p^{6} T^{8} + 2213631 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 89 | \( ( 1 + 337656 T^{2} + 847513884336 T^{4} + 370612343654439478 T^{6} + 847513884336 p^{6} T^{8} + 337656 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
| 97 | \( ( 1 - 2338539 T^{2} + 36704173443 p T^{4} - 3525808839119151986 T^{6} + 36704173443 p^{7} T^{8} - 2338539 p^{12} T^{10} + p^{18} T^{12} )^{2} \) |
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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
−3.07155484374332625446984398775, −3.00456969659347311028396468760, −2.85980055052180926688785635635, −2.79789970124123877594011140111, −2.32220820712239104017467275953, −2.32167307071379774912980542500, −2.30227121543658849724260953572, −2.29403350367368638433149591599, −2.28608284266971660808839862525, −2.24921433313954131992404082849, −1.78734175646132241212026476311, −1.64443947552256986070499710283, −1.56062336692726040960130452546, −1.51160380220314358576760563083, −1.49506721207011258959900672816, −1.45822803550857575388922782796, −1.37730815898735445408375353039, −1.21043996683535349142712322956, −0.889682875219422754524009646616, −0.855566162031657208873835380103, −0.76627010068147160927620732579, −0.40246725091056754425663912969, −0.27963227471846668319581027640, −0.19620708099834105398502462730, −0.11453443179594399573839156868,
0.11453443179594399573839156868, 0.19620708099834105398502462730, 0.27963227471846668319581027640, 0.40246725091056754425663912969, 0.76627010068147160927620732579, 0.855566162031657208873835380103, 0.889682875219422754524009646616, 1.21043996683535349142712322956, 1.37730815898735445408375353039, 1.45822803550857575388922782796, 1.49506721207011258959900672816, 1.51160380220314358576760563083, 1.56062336692726040960130452546, 1.64443947552256986070499710283, 1.78734175646132241212026476311, 2.24921433313954131992404082849, 2.28608284266971660808839862525, 2.29403350367368638433149591599, 2.30227121543658849724260953572, 2.32167307071379774912980542500, 2.32220820712239104017467275953, 2.79789970124123877594011140111, 2.85980055052180926688785635635, 3.00456969659347311028396468760, 3.07155484374332625446984398775
Plot not available for L-functions of degree greater than 10.
