| L(s) = 1 | − 8·5-s + 36·25-s − 16·29-s + 16·49-s − 32·53-s − 16·73-s + 48·97-s − 48·101-s + 32·121-s − 120·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 36/5·25-s − 2.97·29-s + 16/7·49-s − 4.39·53-s − 1.87·73-s + 4.87·97-s − 4.77·101-s + 2.90·121-s − 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8173670772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8173670772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{8} \) |
| good | 7 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 16 T^{2} + 146 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T^{2} + 194 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 12 T^{2} - 26 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 20 T^{2} + 182 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 36 T^{2} + 1606 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T^{2} - 1246 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 2722 T^{4} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 84 T^{2} + 3622 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 16 T^{2} + 3026 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 92 T^{2} + 8534 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 108 T^{2} + 10438 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 164 T^{2} + 13446 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 220 T^{2} + 23318 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 96 T^{2} + 2146 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.69810629251268708048125432013, −3.65397802604053802759616973076, −3.54009010030874009262265038649, −3.34430492201213673616043978915, −3.25412659549503768301830785058, −2.97288406034477227594862936209, −2.96460987065769786169142084343, −2.91180740389960329390749174937, −2.87775203100213665603601178107, −2.62610731972336804167084421317, −2.44148130439187085371275547421, −2.23446985650642023358130340594, −2.23406789782428952933086228177, −1.98944776833456399006603302557, −1.97608923040735250532791765751, −1.68680331926821016828020073144, −1.61491148899301569721629135857, −1.29105222703329476986530154615, −1.16232528327121007233878240809, −1.12515719127505089303640734221, −1.11375305170958920783474797520, −0.58184783937199424772887929117, −0.35687270948545598255833231105, −0.26227075007737036405836084150, −0.24162793023197096361506919913,
0.24162793023197096361506919913, 0.26227075007737036405836084150, 0.35687270948545598255833231105, 0.58184783937199424772887929117, 1.11375305170958920783474797520, 1.12515719127505089303640734221, 1.16232528327121007233878240809, 1.29105222703329476986530154615, 1.61491148899301569721629135857, 1.68680331926821016828020073144, 1.97608923040735250532791765751, 1.98944776833456399006603302557, 2.23406789782428952933086228177, 2.23446985650642023358130340594, 2.44148130439187085371275547421, 2.62610731972336804167084421317, 2.87775203100213665603601178107, 2.91180740389960329390749174937, 2.96460987065769786169142084343, 2.97288406034477227594862936209, 3.25412659549503768301830785058, 3.34430492201213673616043978915, 3.54009010030874009262265038649, 3.65397802604053802759616973076, 3.69810629251268708048125432013
Plot not available for L-functions of degree greater than 10.
