| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s − 8·17-s − 8·19-s + 4·25-s + 4·27-s + 4·29-s − 8·31-s − 8·33-s − 8·37-s − 8·41-s + 8·43-s − 12·45-s + 20·47-s − 6·49-s − 16·51-s − 4·53-s + 16·55-s − 16·57-s + 20·59-s + 8·71-s − 4·73-s + 8·75-s + 12·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s − 1.94·17-s − 1.83·19-s + 4/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.39·33-s − 1.31·37-s − 1.24·41-s + 1.21·43-s − 1.78·45-s + 2.91·47-s − 6/7·49-s − 2.24·51-s − 0.549·53-s + 2.15·55-s − 2.11·57-s + 2.60·59-s + 0.949·71-s − 0.468·73-s + 0.923·75-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64256256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64256256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6753296327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6753296327\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 167 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_4$ | \( 1 - 20 T + 186 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 92 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 132 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 176 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.180103428393529296206690289712, −7.74838956013818935383199512692, −7.27436743993728353419285787670, −7.15995143270332002919940674372, −6.70086571582468789286236077853, −6.64618330549798191451695758216, −5.82324893304271180834739113115, −5.64132984961816535481468629611, −4.97549909455134893697845209193, −4.75270594021407661630319812680, −4.17960514483653865191517050638, −4.17667630494894134572701322074, −3.66860569822804063758825855589, −3.51114989696741108455619800940, −2.83180567757970062369705848841, −2.46718406640213291649726689426, −1.98359931598991980555102016161, −1.98034524652463485856812744997, −0.789276526356961023768938275884, −0.22305366778578173736007627395,
0.22305366778578173736007627395, 0.789276526356961023768938275884, 1.98034524652463485856812744997, 1.98359931598991980555102016161, 2.46718406640213291649726689426, 2.83180567757970062369705848841, 3.51114989696741108455619800940, 3.66860569822804063758825855589, 4.17667630494894134572701322074, 4.17960514483653865191517050638, 4.75270594021407661630319812680, 4.97549909455134893697845209193, 5.64132984961816535481468629611, 5.82324893304271180834739113115, 6.64618330549798191451695758216, 6.70086571582468789286236077853, 7.15995143270332002919940674372, 7.27436743993728353419285787670, 7.74838956013818935383199512692, 8.180103428393529296206690289712
