L(s) = 1 | + 4·3-s + 10·9-s + 8·11-s + 16·19-s − 4·25-s + 20·27-s + 32·33-s + 32·41-s − 16·49-s + 64·57-s + 32·67-s + 16·73-s − 16·75-s + 35·81-s + 8·83-s + 8·89-s + 32·97-s + 80·99-s − 16·107-s + 72·113-s + 20·121-s + 128·123-s + 127-s + 131-s + 137-s + 139-s − 64·147-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 10/3·9-s + 2.41·11-s + 3.67·19-s − 4/5·25-s + 3.84·27-s + 5.57·33-s + 4.99·41-s − 2.28·49-s + 8.47·57-s + 3.90·67-s + 1.87·73-s − 1.84·75-s + 35/9·81-s + 0.878·83-s + 0.847·89-s + 3.24·97-s + 8.04·99-s − 1.54·107-s + 6.77·113-s + 1.81·121-s + 11.5·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.27·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(34.84986251\) |
\(L(\frac12)\) |
\(\approx\) |
\(34.84986251\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 5 | $D_4$ | \( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1254 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 2034 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6918 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 12114 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 2106 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.40022793473364447458911177961, −5.86419575817852579177074794099, −5.79489898903480612273943841283, −5.74899536970872883666464557580, −5.27753602297995164610784448116, −5.03858634664365566238766702091, −4.79178558893748889434439273247, −4.72462497404040423643125040893, −4.57229136410099624432517569983, −3.96595193752890180410464905455, −3.92255310065725778516523564569, −3.86221845182884493905404584267, −3.76554254503579320989267947350, −3.31545313217385080417520618090, −3.16025019486193454359513104479, −3.07709379239301356467124176969, −2.93941496353539956998599925223, −2.21433665071209340817154489239, −2.17643494284857105043127354728, −2.16964616267818509208410897429, −1.86859272527778196603443171995, −1.13720292026675151979891718530, −1.11270505145013229578390795304, −0.963120409678928536648681214448, −0.72924079848520114127583645393,
0.72924079848520114127583645393, 0.963120409678928536648681214448, 1.11270505145013229578390795304, 1.13720292026675151979891718530, 1.86859272527778196603443171995, 2.16964616267818509208410897429, 2.17643494284857105043127354728, 2.21433665071209340817154489239, 2.93941496353539956998599925223, 3.07709379239301356467124176969, 3.16025019486193454359513104479, 3.31545313217385080417520618090, 3.76554254503579320989267947350, 3.86221845182884493905404584267, 3.92255310065725778516523564569, 3.96595193752890180410464905455, 4.57229136410099624432517569983, 4.72462497404040423643125040893, 4.79178558893748889434439273247, 5.03858634664365566238766702091, 5.27753602297995164610784448116, 5.74899536970872883666464557580, 5.79489898903480612273943841283, 5.86419575817852579177074794099, 6.40022793473364447458911177961