| L(s) = 1 | + 16·25-s + 48·41-s − 32·49-s + 16·73-s + 96·89-s + 32·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 16/5·25-s + 7.49·41-s − 4.57·49-s + 1.87·73-s + 10.1·89-s + 3.24·97-s − 0.363·121-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 + 2.46·169-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 + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(34.67098919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(34.67098919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 8 T^{2} + 42 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 16 T^{2} + 138 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 378 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 20 T^{2} + 438 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 56 T^{2} + 1626 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 28 T^{2} + 1494 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 76 T^{2} + 2982 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T^{2} - 714 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 152 T^{2} + 9978 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 104 T^{2} + 8106 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 208 T^{2} + 18042 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 68 T^{2} + 8598 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 56 T^{2} - 1830 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 112 T^{2} + 12714 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 212 T^{2} + 21558 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 8 T - 6 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.22806802983349717416065452140, −3.11415688392008131705966439527, −2.95392231377222884332068811233, −2.87333818320194404337166707481, −2.83755583213773359356486548647, −2.74004129656082703699694734762, −2.72686213218881376643333550508, −2.67345771274800535141410775265, −2.30863437715804307949434986812, −2.24422136812895632321502502677, −1.98718290297241218075238860982, −1.97159010935142060509419199948, −1.92103583695108250414919525920, −1.82897615411541869234339830085, −1.80844668664243555006131034366, −1.66672436096604417287686243087, −1.45186774686040189008803610468, −0.958186791227966243401400480265, −0.910582070331261140799994826724, −0.872247205894755606671260712685, −0.77103196823814794111030101900, −0.76922046417317219103664321421, −0.73671199270499224545945157862, −0.46547243591447458092884040405, −0.23620540459630930340958435454,
0.23620540459630930340958435454, 0.46547243591447458092884040405, 0.73671199270499224545945157862, 0.76922046417317219103664321421, 0.77103196823814794111030101900, 0.872247205894755606671260712685, 0.910582070331261140799994826724, 0.958186791227966243401400480265, 1.45186774686040189008803610468, 1.66672436096604417287686243087, 1.80844668664243555006131034366, 1.82897615411541869234339830085, 1.92103583695108250414919525920, 1.97159010935142060509419199948, 1.98718290297241218075238860982, 2.24422136812895632321502502677, 2.30863437715804307949434986812, 2.67345771274800535141410775265, 2.72686213218881376643333550508, 2.74004129656082703699694734762, 2.83755583213773359356486548647, 2.87333818320194404337166707481, 2.95392231377222884332068811233, 3.11415688392008131705966439527, 3.22806802983349717416065452140
Plot not available for L-functions of degree greater than 10.
