L(s) = 1 | + 48·13-s − 56·25-s + 16·37-s + 152·49-s − 112·61-s − 304·73-s + 112·97-s − 80·109-s + 520·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 840·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 3.69·13-s − 2.23·25-s + 0.432·37-s + 3.10·49-s − 1.83·61-s − 4.16·73-s + 1.15·97-s − 0.733·109-s + 4.29·121-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 + 4.97·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.346303124\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.346303124\) |
\(L(2)\) |
|
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 | \( ( 1 + 28 T^{2} + 22 p^{2} T^{4} + 28 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 76 T^{2} + 5350 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 130 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 12 T + 150 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 868 T^{2} + 341062 T^{4} + 868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 708 T^{2} + 256934 T^{4} - 708 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 196 T^{2} - 348218 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2524 T^{2} + 2963302 T^{4} + 2524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3084 T^{2} + 4152230 T^{4} - 3084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 2518 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3364 T^{2} + 7778182 T^{4} + 3364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 6148 T^{2} + 15928678 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 252 T^{2} + 1517702 T^{4} + 252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4060 T^{2} + 11815462 T^{4} + 4060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8932 T^{2} + 40509862 T^{4} - 8932 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 28 T + 7414 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 676 T^{2} + 21837030 T^{4} - 676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 76 T + 4038 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 17228 T^{2} + 145319334 T^{4} - 17228 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 10948 T^{2} + 121211302 T^{4} - 10948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 11076 T^{2} + 54999110 T^{4} + 11076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 28 T + 15430 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.90909901955442397979229453990, −3.80398957336384929283414269898, −3.78692396507640589353005898672, −3.51190127041677113456299629614, −3.47773142758690205111600761043, −3.32936345853287489197619203496, −3.30232723399910373336582398450, −3.03848017486895691339684665345, −2.97453960962828175796867181515, −2.81393917996652912960625962013, −2.51487126690261679437617563828, −2.37090685504707112094839066227, −2.35209878042078762361228883592, −2.21176566364685470740950149031, −1.89792817871310065152472638681, −1.88438877099315770561840317276, −1.56219875144283458474411231081, −1.54115859415924529939334851710, −1.23169910170895651431909599348, −1.19313536600033497415173466280, −1.06007098367438378461991053494, −0.78138809249653784476987384669, −0.73728069931464738427823381674, −0.21072101565080779048047083565, −0.11295561804468839637723482487,
0.11295561804468839637723482487, 0.21072101565080779048047083565, 0.73728069931464738427823381674, 0.78138809249653784476987384669, 1.06007098367438378461991053494, 1.19313536600033497415173466280, 1.23169910170895651431909599348, 1.54115859415924529939334851710, 1.56219875144283458474411231081, 1.88438877099315770561840317276, 1.89792817871310065152472638681, 2.21176566364685470740950149031, 2.35209878042078762361228883592, 2.37090685504707112094839066227, 2.51487126690261679437617563828, 2.81393917996652912960625962013, 2.97453960962828175796867181515, 3.03848017486895691339684665345, 3.30232723399910373336582398450, 3.32936345853287489197619203496, 3.47773142758690205111600761043, 3.51190127041677113456299629614, 3.78692396507640589353005898672, 3.80398957336384929283414269898, 3.90909901955442397979229453990
Plot not available for L-functions of degree greater than 10.