| L(s) = 1 | + 16·25-s + 40·49-s + 160·73-s + 320·97-s − 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 920·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 0.639·25-s + 0.816·49-s + 2.19·73-s + 3.29·97-s − 5.61·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.44·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.32476062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.32476062\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 4 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 332 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 778 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1010 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 1840 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 4268 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 5810 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 67 | \( ( 1 + 1022 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 4682 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 20 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 2422 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 40 T + p^{2} T^{2} )^{8} \) |
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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
−4.56342282991477425423819386117, −4.24319550087343701880928052982, −4.02092084728768168531163057144, −4.01160025523861825440585401505, −3.79945126930549900899753846462, −3.79691655759987322606253498045, −3.68034939044616539051347417974, −3.64066597204061468515300603701, −3.09137057482780842430955630354, −3.06487555403443334989573372379, −3.01038932760130879744590799196, −2.82868782579970734328022415390, −2.80722130209524658017713091541, −2.44331918331931366998972856504, −2.20221114911522291502177368815, −2.16686380566576402469038978223, −2.14073077571651595316985541677, −1.66396180554149177045964183391, −1.52526443571982145929774445802, −1.37844627845838046588676763718, −1.34153729193530267501157441438, −0.71546652267543075100127325975, −0.62889199641935170677189688218, −0.48550264789576945073043845293, −0.39384308152121996320574497660,
0.39384308152121996320574497660, 0.48550264789576945073043845293, 0.62889199641935170677189688218, 0.71546652267543075100127325975, 1.34153729193530267501157441438, 1.37844627845838046588676763718, 1.52526443571982145929774445802, 1.66396180554149177045964183391, 2.14073077571651595316985541677, 2.16686380566576402469038978223, 2.20221114911522291502177368815, 2.44331918331931366998972856504, 2.80722130209524658017713091541, 2.82868782579970734328022415390, 3.01038932760130879744590799196, 3.06487555403443334989573372379, 3.09137057482780842430955630354, 3.64066597204061468515300603701, 3.68034939044616539051347417974, 3.79691655759987322606253498045, 3.79945126930549900899753846462, 4.01160025523861825440585401505, 4.02092084728768168531163057144, 4.24319550087343701880928052982, 4.56342282991477425423819386117
Plot not available for L-functions of degree greater than 10.
