| L(s) = 1 | + 1.79e5·13-s + 6.83e6·25-s − 1.67e7·37-s + 8.84e7·49-s + 2.73e8·61-s + 1.42e9·73-s + 3.75e9·97-s + 5.85e9·109-s + 2.38e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.23e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.73·13-s + 3.49·25-s − 1.47·37-s + 2.19·49-s + 2.52·61-s + 5.87·73-s + 4.31·97-s + 3.97·109-s + 1.01·121-s − 2.11·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(13.86274799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.86274799\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 136648 p^{2} T^{2} + p^{18} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6319682 p T^{2} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1192153898 T^{2} + p^{18} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 44764 T + p^{9} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 207620627056 T^{2} + p^{18} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 250921932518 T^{2} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 3596397273646 T^{2} + p^{18} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 28639066234840 T^{2} + p^{18} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 7050250770302 T^{2} + p^{18} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4199038 T + p^{9} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 548528412682960 T^{2} + p^{18} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 155801900708246 T^{2} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2022729400421854 T^{2} + p^{18} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4945805394752824 T^{2} + p^{18} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 14860468143977398 T^{2} + p^{18} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 68255030 T + p^{9} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 810090974107202 p T^{2} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 44821764380332658 T^{2} + p^{18} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 356245936 T + p^{9} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 227290249101503198 T^{2} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1134302306330086 T^{2} + p^{18} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 51248190888147040 T^{2} + p^{18} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 939817672 T + p^{9} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.140774517784159189313165355227, −7.43734730904054326603548654875, −7.08107104049448492167862324846, −6.97690971102585619641328614550, −6.92597245295702107045847571115, −6.31244261495426122413558131557, −6.22292932936335317110504122898, −5.90899550098601811547083854572, −5.57217435858432634196624888594, −5.01187947145128865530422149659, −4.95013979322211015826009209754, −4.87038721098177135365663690843, −4.35082860738156719498734363281, −3.63946465047867279011217111838, −3.62124164193539430988170425334, −3.59461337721385899599986035993, −3.15135015142422543107253077780, −2.46626954910081531061952643545, −2.42279581469477190011568728466, −2.02624472786866293782492374718, −1.63590846854082615969028631655, −1.01658227209301594995301496657, −0.867011001347664704307119318529, −0.74432086730645040526746846818, −0.44747846105583180015984842995,
0.44747846105583180015984842995, 0.74432086730645040526746846818, 0.867011001347664704307119318529, 1.01658227209301594995301496657, 1.63590846854082615969028631655, 2.02624472786866293782492374718, 2.42279581469477190011568728466, 2.46626954910081531061952643545, 3.15135015142422543107253077780, 3.59461337721385899599986035993, 3.62124164193539430988170425334, 3.63946465047867279011217111838, 4.35082860738156719498734363281, 4.87038721098177135365663690843, 4.95013979322211015826009209754, 5.01187947145128865530422149659, 5.57217435858432634196624888594, 5.90899550098601811547083854572, 6.22292932936335317110504122898, 6.31244261495426122413558131557, 6.92597245295702107045847571115, 6.97690971102585619641328614550, 7.08107104049448492167862324846, 7.43734730904054326603548654875, 8.140774517784159189313165355227
