| L(s) = 1 | + 12·13-s − 16-s + 12·17-s − 12·19-s + 12·23-s − 8·25-s − 12·29-s − 12·31-s − 8·37-s − 12·43-s − 12·47-s − 24·53-s − 16·67-s − 12·71-s + 24·73-s + 24·79-s + 12·83-s + 8·97-s + 8·103-s − 24·107-s − 24·109-s + 12·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3.32·13-s − 1/4·16-s + 2.91·17-s − 2.75·19-s + 2.50·23-s − 8/5·25-s − 2.22·29-s − 2.15·31-s − 1.31·37-s − 1.82·43-s − 1.75·47-s − 3.29·53-s − 1.95·67-s − 1.42·71-s + 2.80·73-s + 2.70·79-s + 1.31·83-s + 0.812·97-s + 0.788·103-s − 2.32·107-s − 2.29·109-s + 1.12·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288600739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288600739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - 73 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 312 T^{3} + 1271 T^{4} - 312 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 288 T^{3} + 2591 T^{4} + 288 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1554 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 926 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 732 T^{3} + 7246 T^{4} + 732 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2400 T^{3} + 17791 T^{4} + 2400 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 15819 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 780 T^{3} + 8126 T^{4} - 780 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 194 T^{2} + 22659 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 552 T^{3} + 8738 T^{4} - 552 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.20318001785637937797074477512, −6.88768044797210332637749559011, −6.58641585741204696779464791705, −6.54801107782220235094223454908, −6.20223505446603838990922612718, −6.02455227904572526377127659458, −5.89128807894156093244216589367, −5.56263531152455894216639208597, −5.35134827074287669175058900577, −5.08241740602271138130114865723, −5.06336158352744216836768588979, −4.52652664372856828248971071489, −4.34467932009910092448122476445, −3.82535047313831221954704385688, −3.73117229756279963104904124963, −3.63340420371898848519540910653, −3.26415354277592384756977669030, −3.12755128792556715791180606669, −3.10638228846982929432196863281, −2.04386976869755305858852577647, −1.95482911453658214123800741662, −1.75451436946717394040940842887, −1.39743615772228005275759509168, −1.11390340290484634639614836049, −0.25610839637164337986370011531,
0.25610839637164337986370011531, 1.11390340290484634639614836049, 1.39743615772228005275759509168, 1.75451436946717394040940842887, 1.95482911453658214123800741662, 2.04386976869755305858852577647, 3.10638228846982929432196863281, 3.12755128792556715791180606669, 3.26415354277592384756977669030, 3.63340420371898848519540910653, 3.73117229756279963104904124963, 3.82535047313831221954704385688, 4.34467932009910092448122476445, 4.52652664372856828248971071489, 5.06336158352744216836768588979, 5.08241740602271138130114865723, 5.35134827074287669175058900577, 5.56263531152455894216639208597, 5.89128807894156093244216589367, 6.02455227904572526377127659458, 6.20223505446603838990922612718, 6.54801107782220235094223454908, 6.58641585741204696779464791705, 6.88768044797210332637749559011, 7.20318001785637937797074477512
