| L(s) = 1 | − 8·5-s + 4·9-s + 20·25-s + 24·29-s − 32·45-s + 28·49-s − 24·53-s + 16·73-s + 7·81-s + 32·97-s + 8·101-s + 40·121-s + 40·125-s + 127-s + 131-s + 137-s + 139-s − 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 4/3·9-s + 4·25-s + 4.45·29-s − 4.77·45-s + 4·49-s − 3.29·53-s + 1.87·73-s + 7/9·81-s + 3.24·97-s + 0.796·101-s + 3.63·121-s + 3.57·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122911566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122911566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.89245776735874364882668999942, −6.54103146430372627305163242134, −6.30547484269499627215333079340, −6.10567526688660258450194525685, −6.09534328028004569480748805490, −5.80840638563320594206726051014, −5.18273601483575999237607820254, −4.96466610959582757898606594096, −4.95752756360168355056019601466, −4.51707241191267948528500010703, −4.50577694903578521794119785075, −4.33962988997374348519720475726, −4.04830425408162078324773628928, −3.85153840393882762122689353505, −3.59201855669898587837463098868, −3.41556007538124744767088719209, −3.24698712704535614838601604708, −2.93516745779505436392770106098, −2.47210091461415911988970203228, −2.24490596914716846360011966499, −1.99878694112364934126706787079, −1.33705425262239648606468623759, −0.952028810022915412685974956980, −0.77929826956044593007155679201, −0.30564412025589270979591144096,
0.30564412025589270979591144096, 0.77929826956044593007155679201, 0.952028810022915412685974956980, 1.33705425262239648606468623759, 1.99878694112364934126706787079, 2.24490596914716846360011966499, 2.47210091461415911988970203228, 2.93516745779505436392770106098, 3.24698712704535614838601604708, 3.41556007538124744767088719209, 3.59201855669898587837463098868, 3.85153840393882762122689353505, 4.04830425408162078324773628928, 4.33962988997374348519720475726, 4.50577694903578521794119785075, 4.51707241191267948528500010703, 4.95752756360168355056019601466, 4.96466610959582757898606594096, 5.18273601483575999237607820254, 5.80840638563320594206726051014, 6.09534328028004569480748805490, 6.10567526688660258450194525685, 6.30547484269499627215333079340, 6.54103146430372627305163242134, 6.89245776735874364882668999942
