| L(s) = 1 | − 324·9-s + 4.08e3·17-s + 9.73e3·25-s + 5.87e4·41-s − 8.92e4·49-s + 2.48e5·73-s + 6.56e4·81-s − 4.68e4·89-s − 2.62e5·97-s + 1.55e5·113-s + 6.88e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.32e6·153-s + 157-s + 163-s + 167-s + 9.90e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 3.42·17-s + 3.11·25-s + 5.45·41-s − 5.30·49-s + 5.46·73-s + 10/9·81-s − 0.626·89-s − 2.83·97-s + 1.14·113-s + 4.27·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 4.56·153-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2.66·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.097948429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.097948429\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{4} T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4868 T^{2} + 21308406 T^{4} - 4868 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 7 | \( ( 1 + 44620 T^{2} + 996331398 T^{4} + 44620 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 - 344140 T^{2} + 59503437654 T^{4} - 344140 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 13 | \( ( 1 - 495028 T^{2} + 237600644406 T^{4} - 495028 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 - 60 p T + 2726566 T^{2} - 60 p^{6} T^{3} + p^{10} T^{4} )^{4} \) |
| 19 | \( ( 1 - 9090796 T^{2} + 32788406214006 T^{4} - 9090796 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 + 17342108 T^{2} + 146667184039014 T^{4} + 17342108 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 31556324 T^{2} + 1066488933141654 T^{4} - 31556324 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 31 | \( ( 1 + 57418156 T^{2} + 2460980676974694 T^{4} + 57418156 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 37 | \( ( 1 - 121969876 T^{2} + 12520504776830550 T^{4} - 121969876 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 41 | \( ( 1 - 14676 T + 282199414 T^{2} - 14676 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 43 | \( ( 1 - 260676556 T^{2} + 45997306791223254 T^{4} - 260676556 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 47 | \( ( 1 + 168452732 T^{2} - 10462142247546 p^{2} T^{4} + 168452732 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 - 447608516 T^{2} + 39456592299340854 T^{4} - 447608516 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 59 | \( ( 1 - 2805107212 T^{2} + 2988934357466540310 T^{4} - 2805107212 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 1367750068 T^{2} + 1891349188870635510 T^{4} - 1367750068 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4222517932 T^{2} + 7856253565223085366 T^{4} - 4222517932 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3439173028 T^{2} + 9180697944544976166 T^{4} - 3439173028 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 - 62204 T + 2881288662 T^{2} - 62204 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8138966380 T^{2} + 34002365860721433894 T^{4} + 8138966380 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 83 | \( ( 1 - 15210081772 T^{2} + 88864509019951371894 T^{4} - 15210081772 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 + 11700 T - 4027669994 T^{2} + 11700 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 97 | \( ( 1 + 65636 T + 2616923526 T^{2} + 65636 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
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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
−4.09594691057521217344829889202, −4.06394842871701560071124065388, −3.65747085288705921388645248457, −3.63227302879965273678724203113, −3.41778995087764351599293821878, −3.33564628942146320765617008258, −3.25777360646729844245885341406, −2.94811109599433666989977978818, −2.94091948104527669269082523960, −2.86565981708467151568466435058, −2.73867863560066430829653904233, −2.56727115799450821158524960026, −2.13789885978684166604100076769, −2.07184122006874892776214492451, −2.04779106494705187123738975549, −1.72575500285745857147937239046, −1.56628596980090311188210689701, −1.32636240211687958432862535136, −1.10833867997562345180357419400, −0.910984138239736715890876969824, −0.811263759657787389278640853452, −0.77595525207509218012553893864, −0.57221400735010485471802966178, −0.47381967834660392936066254482, −0.06694256902013980415240324027,
0.06694256902013980415240324027, 0.47381967834660392936066254482, 0.57221400735010485471802966178, 0.77595525207509218012553893864, 0.811263759657787389278640853452, 0.910984138239736715890876969824, 1.10833867997562345180357419400, 1.32636240211687958432862535136, 1.56628596980090311188210689701, 1.72575500285745857147937239046, 2.04779106494705187123738975549, 2.07184122006874892776214492451, 2.13789885978684166604100076769, 2.56727115799450821158524960026, 2.73867863560066430829653904233, 2.86565981708467151568466435058, 2.94091948104527669269082523960, 2.94811109599433666989977978818, 3.25777360646729844245885341406, 3.33564628942146320765617008258, 3.41778995087764351599293821878, 3.63227302879965273678724203113, 3.65747085288705921388645248457, 4.06394842871701560071124065388, 4.09594691057521217344829889202
Plot not available for L-functions of degree greater than 10.
