| L(s) = 1 | − 6·4-s − 4·9-s + 13·16-s + 48·19-s + 24·36-s − 56·49-s + 72·59-s − 20·64-s − 288·76-s + 2·81-s − 16·89-s − 8·101-s − 68·121-s + 127-s + 131-s + 137-s + 139-s − 52·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 112·169-s − 192·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3·4-s − 4/3·9-s + 13/4·16-s + 11.0·19-s + 4·36-s − 8·49-s + 9.37·59-s − 5/2·64-s − 33.0·76-s + 2/9·81-s − 1.69·89-s − 0.796·101-s − 6.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8.61·169-s − 14.6·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.240943699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.240943699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( ( 1 + 3 T^{2} + 7 T^{4} + 19 T^{6} + 7 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 3 | \( ( 1 + 2 T^{2} + 5 T^{4} + 10 p T^{6} + 5 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 + 4 p T^{2} + 55 p T^{4} + 3330 T^{6} + 55 p^{3} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 34 T^{2} + 423 T^{4} + 3738 T^{6} + 423 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 56 T^{2} + 1487 T^{4} + 24028 T^{6} + 1487 p^{2} T^{8} + 56 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 4 T + p T^{2} )^{12} \) |
| 23 | \( ( 1 + 66 T^{2} + 1681 T^{4} + 31782 T^{6} + 1681 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 3963 T^{4} + 141760 T^{6} + 3963 p^{2} T^{8} + 80 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 120 T^{2} + 7483 T^{4} + 286390 T^{6} + 7483 p^{2} T^{8} + 120 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 184 T^{2} + 15091 T^{4} + 714048 T^{6} + 15091 p^{2} T^{8} + 184 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 + 94 T^{2} + 4463 T^{4} + 185188 T^{6} + 4463 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 192 T^{2} + 16987 T^{4} + 908436 T^{6} + 16987 p^{2} T^{8} + 192 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 200 T^{2} + 19687 T^{4} + 1160700 T^{6} + 19687 p^{2} T^{8} + 200 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 162 T^{2} + 16447 T^{4} + 1020796 T^{6} + 16447 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 6 T + p T^{2} )^{12} \) |
| 61 | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{6} \) |
| 67 | \( ( 1 + 188 T^{2} + 21487 T^{4} + 1699284 T^{6} + 21487 p^{2} T^{8} + 188 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 294 T^{2} + 42523 T^{4} + 3776858 T^{6} + 42523 p^{2} T^{8} + 294 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 286 T^{2} + 36127 T^{4} + 3003876 T^{6} + 36127 p^{2} T^{8} + 286 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 300 T^{2} + 46983 T^{4} + 4550550 T^{6} + 46983 p^{2} T^{8} + 300 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 440 T^{2} + 84927 T^{4} + 9176220 T^{6} + 84927 p^{2} T^{8} + 440 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 4 T + 47 T^{2} - 878 T^{3} + 47 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 + 64 T^{2} + 19851 T^{4} + 827088 T^{6} + 19851 p^{2} T^{8} + 64 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.30510210110262100409794332477, −2.29651837243719190624755209784, −2.10708300577847334162384367574, −2.03605495934795650364679780271, −1.93571347396201792911691822694, −1.81848437412217820714349530520, −1.73743215092382650266916465921, −1.72124512476517175309303494084, −1.67837862507464272989938044452, −1.51866301601796093418260807194, −1.45153724093597613979517661278, −1.41905982344363651498736918882, −1.33173619821029701821130344293, −1.27050768474801202204607965892, −1.00414958506121348175488206530, −0.935498091043930806611411516855, −0.881486303552853429381122411462, −0.855284509943932983493051417777, −0.807018567883485157409996510420, −0.54589744580402689748021805651, −0.52773156622842347624090280356, −0.48007195547037886556522388607, −0.43769093035855149116036977311, −0.37359966226559406806715464679, −0.089898234606209017699545171593,
0.089898234606209017699545171593, 0.37359966226559406806715464679, 0.43769093035855149116036977311, 0.48007195547037886556522388607, 0.52773156622842347624090280356, 0.54589744580402689748021805651, 0.807018567883485157409996510420, 0.855284509943932983493051417777, 0.881486303552853429381122411462, 0.935498091043930806611411516855, 1.00414958506121348175488206530, 1.27050768474801202204607965892, 1.33173619821029701821130344293, 1.41905982344363651498736918882, 1.45153724093597613979517661278, 1.51866301601796093418260807194, 1.67837862507464272989938044452, 1.72124512476517175309303494084, 1.73743215092382650266916465921, 1.81848437412217820714349530520, 1.93571347396201792911691822694, 2.03605495934795650364679780271, 2.10708300577847334162384367574, 2.29651837243719190624755209784, 2.30510210110262100409794332477
Plot not available for L-functions of degree greater than 10.
