| L(s) = 1 | − 4·13-s − 24·17-s + 24·29-s − 20·37-s + 24·49-s − 24·53-s − 20·61-s − 48·97-s − 24·101-s + 52·109-s − 24·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 5.82·17-s + 4.45·29-s − 3.28·37-s + 24/7·49-s − 3.29·53-s − 2.56·61-s − 4.87·97-s − 2.38·101-s + 4.98·109-s − 2.25·113-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6784035427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6784035427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.38600644488437615369102147000, −6.30503206418632232052932633575, −6.23288245415410835256997254398, −5.78839142806257455153022381807, −5.55298413978185410955867840963, −5.17469720094715453437008499386, −5.13905627158435542365683695826, −4.70685389115854170088387727069, −4.64499638070020071096314864984, −4.51067837790152683634900013540, −4.49445709084411454844533274266, −4.16797541630181040620123468212, −3.95841575990851241331003382181, −3.53393521309294082321785235390, −3.37629555557710574972288974723, −2.88665714932374340011344779327, −2.62008963743873140182542144088, −2.58104023488496124966257506820, −2.53127196562431032126093725849, −2.07623991382578500499876583397, −1.82985133504901376573277304707, −1.37926782333916385170133901781, −1.32019245594204322139546197352, −0.36781669003870647882598076445, −0.25745859414512768852423462568,
0.25745859414512768852423462568, 0.36781669003870647882598076445, 1.32019245594204322139546197352, 1.37926782333916385170133901781, 1.82985133504901376573277304707, 2.07623991382578500499876583397, 2.53127196562431032126093725849, 2.58104023488496124966257506820, 2.62008963743873140182542144088, 2.88665714932374340011344779327, 3.37629555557710574972288974723, 3.53393521309294082321785235390, 3.95841575990851241331003382181, 4.16797541630181040620123468212, 4.49445709084411454844533274266, 4.51067837790152683634900013540, 4.64499638070020071096314864984, 4.70685389115854170088387727069, 5.13905627158435542365683695826, 5.17469720094715453437008499386, 5.55298413978185410955867840963, 5.78839142806257455153022381807, 6.23288245415410835256997254398, 6.30503206418632232052932633575, 6.38600644488437615369102147000
