| L(s) = 1 | − 16·13-s + 8·25-s + 32·37-s − 4·49-s + 32·73-s + 64·97-s − 64·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 8/5·25-s + 5.26·37-s − 4/7·49-s + 3.74·73-s + 6.49·97-s − 6.13·109-s − 2.90·121-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 + 3.07·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.597139073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.597139073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 16 T^{2} + 114 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T^{2} + 1014 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 64 T^{2} + 1890 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2514 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 52 T^{2} + 3606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 44 T^{2} + 2550 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 160 T^{2} + 14754 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 116 T^{2} + 10230 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 340 T^{2} + 44694 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.42568263243295025944483309618, −3.37581576242491460827708332367, −3.31947394032537605375467319529, −2.93046905773789591252870197995, −2.90453478141764417338611059221, −2.88513578359844996642913323979, −2.64979457723044386617528102893, −2.62756090221808372127290672174, −2.46499565682000494933980818291, −2.40588076424096118894316504360, −2.40182569891396509306676571593, −2.32011793205413141339681012209, −2.30487202291604788152006065815, −2.00040718461364045831048168383, −1.79028862109482385363615671781, −1.57018560794972513846132592052, −1.48905002221123286265998450509, −1.35873983513891502876849621757, −1.30049939429184562092535839994, −0.898709183509367517827594519479, −0.856363261104878066327038119864, −0.69281928844263599315628919473, −0.53242413912043393420610075780, −0.37708266652230527765641067874, −0.13198158464834490890815825075,
0.13198158464834490890815825075, 0.37708266652230527765641067874, 0.53242413912043393420610075780, 0.69281928844263599315628919473, 0.856363261104878066327038119864, 0.898709183509367517827594519479, 1.30049939429184562092535839994, 1.35873983513891502876849621757, 1.48905002221123286265998450509, 1.57018560794972513846132592052, 1.79028862109482385363615671781, 2.00040718461364045831048168383, 2.30487202291604788152006065815, 2.32011793205413141339681012209, 2.40182569891396509306676571593, 2.40588076424096118894316504360, 2.46499565682000494933980818291, 2.62756090221808372127290672174, 2.64979457723044386617528102893, 2.88513578359844996642913323979, 2.90453478141764417338611059221, 2.93046905773789591252870197995, 3.31947394032537605375467319529, 3.37581576242491460827708332367, 3.42568263243295025944483309618
Plot not available for L-functions of degree greater than 10.
