| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 2·7-s + 4·8-s − 3·9-s − 4·10-s − 4·14-s + 5·16-s − 4·17-s − 6·18-s − 2·19-s − 6·20-s + 3·25-s − 6·28-s + 2·29-s − 10·31-s + 6·32-s − 8·34-s + 4·35-s − 9·36-s − 16·37-s − 4·38-s − 8·40-s + 4·41-s − 8·43-s + 6·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s − 9-s − 1.26·10-s − 1.06·14-s + 5/4·16-s − 0.970·17-s − 1.41·18-s − 0.458·19-s − 1.34·20-s + 3/5·25-s − 1.13·28-s + 0.371·29-s − 1.79·31-s + 1.06·32-s − 1.37·34-s + 0.676·35-s − 3/2·36-s − 2.63·37-s − 0.648·38-s − 1.26·40-s + 0.624·41-s − 1.21·43-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 92 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 71 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 131 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 188 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 155 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.889380013817030033532315537819, −7.50317992606350628817361073654, −7.17654556680826049700471754824, −6.80867175334490204405230624948, −6.45787930808684760345840156059, −6.40969332465575892570866478318, −5.59683468868526369185990195397, −5.48998804059451586097240085382, −5.09934204181527998326938676943, −4.81354510472254401576218933202, −4.02089084361273029699273033823, −4.00068800680000580385973145462, −3.56397338066053608267445079382, −3.29494968027055432555737442050, −2.59117730137126331435231253087, −2.50811855396218150657590962044, −1.85762107957747586034295888855, −1.21408934674687223065816340275, 0, 0,
1.21408934674687223065816340275, 1.85762107957747586034295888855, 2.50811855396218150657590962044, 2.59117730137126331435231253087, 3.29494968027055432555737442050, 3.56397338066053608267445079382, 4.00068800680000580385973145462, 4.02089084361273029699273033823, 4.81354510472254401576218933202, 5.09934204181527998326938676943, 5.48998804059451586097240085382, 5.59683468868526369185990195397, 6.40969332465575892570866478318, 6.45787930808684760345840156059, 6.80867175334490204405230624948, 7.17654556680826049700471754824, 7.50317992606350628817361073654, 7.889380013817030033532315537819
