| L(s) = 1 | − 22·4-s − 288·9-s − 376·11-s − 297·16-s + 1.75e3·25-s − 64·29-s + 6.33e3·36-s + 8.27e3·44-s − 3.91e3·49-s + 1.08e4·64-s − 1.80e4·71-s − 8.62e3·79-s + 2.55e4·81-s + 1.08e5·99-s − 3.85e4·100-s − 1.42e4·109-s + 1.40e3·116-s − 7.04e3·121-s + 127-s + 131-s + 137-s + 139-s + 8.55e4·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.37·4-s − 3.55·9-s − 3.10·11-s − 1.16·16-s + 14/5·25-s − 0.0760·29-s + 44/9·36-s + 4.27·44-s − 1.63·49-s + 2.64·64-s − 3.57·71-s − 1.38·79-s + 3.90·81-s + 11.0·99-s − 3.84·100-s − 1.20·109-s + 0.104·116-s − 0.480·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 33/8·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01310578511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01310578511\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 14 p^{3} T^{2} + 61026 p^{2} T^{4} - 14 p^{11} T^{6} + p^{16} T^{8} \) |
| 7 | \( 1 + 3914 T^{2} + 188274 p^{2} T^{4} + 3914 p^{8} T^{6} + p^{16} T^{8} \) |
| good | 2 | \( ( 1 + 11 T^{2} + 165 p T^{4} + 11 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 3 | \( ( 1 + 8 p^{2} T^{2} + p^{8} T^{4} )^{4} \) |
| 11 | \( ( 1 + 94 T + 23850 T^{2} + 94 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 13 | \( ( 1 + 61994 T^{2} + 2055686226 T^{4} + 61994 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6652 p T^{2} + 13684253046 T^{4} + 6652 p^{9} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 - 173434 T^{2} + 21557516946 T^{4} - 173434 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 23 | \( ( 1 - 107434 T^{2} - 95560036110 T^{4} - 107434 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 + 16 T + 1411230 T^{2} + 16 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1212004 T^{2} + 1906616498886 T^{4} - 1212004 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4830304 T^{2} + 11331475857150 T^{4} - 4830304 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 - 10187224 T^{2} + 41665023662286 T^{4} - 10187224 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4957654 T^{2} + 29482221016530 T^{4} - 4957654 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 18619394 T^{2} + 134260132274706 T^{4} + 18619394 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 - 30454816 T^{2} + 356109945049086 T^{4} - 30454816 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 23023594 T^{2} + 304356594784626 T^{4} - 23023594 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10515094 T^{2} + 407168086576146 T^{4} - 10515094 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 - 41903254 T^{2} + 1234550386896210 T^{4} - 41903254 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4504 T + 48708930 T^{2} + 4504 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 73 | \( ( 1 + 39860804 T^{2} + 919442495343366 T^{4} + 39860804 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 2156 T + 70229250 T^{2} + 2156 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 83 | \( ( 1 + 189429584 T^{2} + 13475437948399746 T^{4} + 189429584 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 81241564 T^{2} + 2338847761996086 T^{4} - 81241564 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 193993724 T^{2} + 18673991144517366 T^{4} + 193993724 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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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
−7.32288903750104086214190115331, −6.98454700235663065169350223685, −6.94580291500696261808412415664, −6.72985181107401930315769546944, −6.49734034705987263069442121363, −5.92367791064675078628855811911, −5.86901721906674966994327014228, −5.71660973444130045559930456240, −5.66806535565878929366672364506, −5.34940426863180357246323810730, −5.08021399825307536930319945838, −4.86484273250298672419406482733, −4.66170814000672037618657579480, −4.52744882253376894431762094793, −4.38681058252662227993113161865, −3.77265373106266854191733007917, −3.29111042483137079434970634427, −3.12588661669237353532786683790, −2.74955382712773363476653049562, −2.74375893231516884113611343042, −2.59016448918214736749619526038, −2.01888583232201417016074087974, −1.21895894398514212463003854258, −0.32141121756519341972011670848, −0.06318840799050324194359853591,
0.06318840799050324194359853591, 0.32141121756519341972011670848, 1.21895894398514212463003854258, 2.01888583232201417016074087974, 2.59016448918214736749619526038, 2.74375893231516884113611343042, 2.74955382712773363476653049562, 3.12588661669237353532786683790, 3.29111042483137079434970634427, 3.77265373106266854191733007917, 4.38681058252662227993113161865, 4.52744882253376894431762094793, 4.66170814000672037618657579480, 4.86484273250298672419406482733, 5.08021399825307536930319945838, 5.34940426863180357246323810730, 5.66806535565878929366672364506, 5.71660973444130045559930456240, 5.86901721906674966994327014228, 5.92367791064675078628855811911, 6.49734034705987263069442121363, 6.72985181107401930315769546944, 6.94580291500696261808412415664, 6.98454700235663065169350223685, 7.32288903750104086214190115331
Plot not available for L-functions of degree greater than 10.
