| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·6-s + 3·9-s − 4·11-s + 2·12-s − 2·13-s + 16-s + 4·17-s − 6·18-s + 8·22-s − 8·23-s − 2·25-s + 4·26-s + 4·27-s + 4·29-s − 8·31-s + 2·32-s − 8·33-s − 8·34-s + 3·36-s − 4·37-s − 4·39-s + 16·41-s + 8·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s + 9-s − 1.20·11-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 1.70·22-s − 1.66·23-s − 2/5·25-s + 0.784·26-s + 0.769·27-s + 0.742·29-s − 1.43·31-s + 0.353·32-s − 1.39·33-s − 1.37·34-s + 1/2·36-s − 0.657·37-s − 0.640·39-s + 2.49·41-s + 1.21·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3820404688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3820404688\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.47979635758244387345815012710, −16.04950298307033131008500397708, −15.65522767945660065950396188802, −14.79138962396656964896705769639, −14.25172602711280113378671227511, −13.94722132875602897823387217019, −12.77014286161494746770905877170, −12.75072538387550213063268470792, −11.73521024606481280324460127322, −10.76283283523425897955276804858, −10.09067269620032594182897164525, −9.688539265815594496355075871852, −9.187226201934996997898965277831, −8.367796056521928312460730790959, −7.79055936858715051043363136838, −7.60867930956735318051034647701, −6.21031444191802945622313938697, −5.06473968936367901043856637263, −3.71045824363177121690250957726, −2.40208152373406171820840496566,
2.40208152373406171820840496566, 3.71045824363177121690250957726, 5.06473968936367901043856637263, 6.21031444191802945622313938697, 7.60867930956735318051034647701, 7.79055936858715051043363136838, 8.367796056521928312460730790959, 9.187226201934996997898965277831, 9.688539265815594496355075871852, 10.09067269620032594182897164525, 10.76283283523425897955276804858, 11.73521024606481280324460127322, 12.75072538387550213063268470792, 12.77014286161494746770905877170, 13.94722132875602897823387217019, 14.25172602711280113378671227511, 14.79138962396656964896705769639, 15.65522767945660065950396188802, 16.04950298307033131008500397708, 16.47979635758244387345815012710
