| L(s) = 1 | + 88·25-s + 168·49-s + 208·73-s + 144·97-s − 840·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.28e3·169-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 | + 3.51·25-s + 24/7·49-s + 2.84·73-s + 1.48·97-s − 6.94·121-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 + 7.62·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.631723622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631723622\) |
| \(L(2)\) |
|
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 | \( ( 1 - 22 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 - 6 p T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 174 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 546 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 - 1654 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1418 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1954 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 670 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 2802 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 - 3350 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 3406 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 5394 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 1890 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 26 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 + 3702 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 14050 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18 T + p^{2} T^{2} )^{8} \) |
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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
−4.04231032551772626159569517793, −3.99977206453199312679625293246, −3.57954861806703155210681837344, −3.57445486295418085256722010699, −3.36879696503602010335270439915, −3.19064071812404276465968768338, −3.17348423941846836989068791107, −3.07998819732355007396660532401, −3.06523072752862945526910690953, −2.64091865129569971588356299818, −2.59286714672232071685918215017, −2.46827739988855517484640408431, −2.22961033412952154117903460327, −2.20476986563474091561823070148, −2.13749406137433368059297000140, −2.01177067866957031477257909464, −1.52540446504171779292672882311, −1.45558348571195851950652241976, −1.25243428382924088574053173678, −1.12808331566065443737404782704, −1.02864294030720952459993727424, −0.71396711984107043857150220849, −0.60878451096137614977799644058, −0.50650366440550799180410077335, −0.07095314517901251428798162063,
0.07095314517901251428798162063, 0.50650366440550799180410077335, 0.60878451096137614977799644058, 0.71396711984107043857150220849, 1.02864294030720952459993727424, 1.12808331566065443737404782704, 1.25243428382924088574053173678, 1.45558348571195851950652241976, 1.52540446504171779292672882311, 2.01177067866957031477257909464, 2.13749406137433368059297000140, 2.20476986563474091561823070148, 2.22961033412952154117903460327, 2.46827739988855517484640408431, 2.59286714672232071685918215017, 2.64091865129569971588356299818, 3.06523072752862945526910690953, 3.07998819732355007396660532401, 3.17348423941846836989068791107, 3.19064071812404276465968768338, 3.36879696503602010335270439915, 3.57445486295418085256722010699, 3.57954861806703155210681837344, 3.99977206453199312679625293246, 4.04231032551772626159569517793
Plot not available for L-functions of degree greater than 10.
