| L(s) = 1 | + 4·4-s + 12·11-s + 7·16-s + 20·19-s + 2·25-s + 4·29-s − 28·31-s + 48·44-s + 16·49-s − 16·61-s + 8·64-s + 80·76-s − 4·79-s + 8·100-s + 24·101-s − 16·109-s + 16·116-s + 52·121-s − 112·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3.61·11-s + 7/4·16-s + 4.58·19-s + 2/5·25-s + 0.742·29-s − 5.02·31-s + 7.23·44-s + 16/7·49-s − 2.04·61-s + 64-s + 9.17·76-s − 0.450·79-s + 4/5·100-s + 2.38·101-s − 1.53·109-s + 1.48·116-s + 4.72·121-s − 10.0·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.51167909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.51167909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1266 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2034 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3906 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 7686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 22866 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 232 T^{2} + 30546 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.97585230613869630036786455512, −6.87804770674786852918725352498, −6.47472115167800355813335429708, −6.24941387113872404598489617600, −6.15672906155048793008843565666, −5.74185140181659740906634609545, −5.53363029759356559362906251168, −5.50092530078822640510322272217, −5.33530041851491305793208089580, −5.03275282368382959792285631489, −4.44578655545895883651390693756, −4.33378141021422295472863150773, −4.19895240962364984137106134266, −3.68976225311142910022509208897, −3.49134581590667326319321262017, −3.35889720522888311486638212939, −3.22672639264062564852996615128, −3.09204436440169087042401383369, −2.36804620841260282479037982889, −2.28253277097415837472278729178, −1.89511295743707659131701092410, −1.45732607131558302754149466085, −1.35534130192680599769842209377, −1.18459633480461727541049630143, −0.66200327723905549043118126433,
0.66200327723905549043118126433, 1.18459633480461727541049630143, 1.35534130192680599769842209377, 1.45732607131558302754149466085, 1.89511295743707659131701092410, 2.28253277097415837472278729178, 2.36804620841260282479037982889, 3.09204436440169087042401383369, 3.22672639264062564852996615128, 3.35889720522888311486638212939, 3.49134581590667326319321262017, 3.68976225311142910022509208897, 4.19895240962364984137106134266, 4.33378141021422295472863150773, 4.44578655545895883651390693756, 5.03275282368382959792285631489, 5.33530041851491305793208089580, 5.50092530078822640510322272217, 5.53363029759356559362906251168, 5.74185140181659740906634609545, 6.15672906155048793008843565666, 6.24941387113872404598489617600, 6.47472115167800355813335429708, 6.87804770674786852918725352498, 6.97585230613869630036786455512
