L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 2·11-s + 12-s − 2·13-s + 16-s + 3·17-s + 18-s + 19-s + 2·22-s − 23-s + 24-s − 5·25-s − 2·26-s + 27-s + 2·29-s − 7·31-s + 32-s + 2·33-s + 3·34-s + 36-s − 2·37-s + 38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 1.25·31-s + 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s − 0.328·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.342976749\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.342976749\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 100 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 80 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 160 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−15.0854116851, −14.6786987313, −14.3456170058, −13.7650490612, −13.4694045228, −13.0649631898, −12.2205782583, −12.0533151356, −11.7488827876, −10.9693499117, −10.3510873157, −10.0227907625, −9.44726304195, −8.83008456972, −8.38016342819, −7.58590889642, −7.28949475388, −6.69056689649, −5.93359745427, −5.38685204147, −4.74588437349, −3.85402390578, −3.52135541382, −2.54009404536, −1.63623526929,
1.63623526929, 2.54009404536, 3.52135541382, 3.85402390578, 4.74588437349, 5.38685204147, 5.93359745427, 6.69056689649, 7.28949475388, 7.58590889642, 8.38016342819, 8.83008456972, 9.44726304195, 10.0227907625, 10.3510873157, 10.9693499117, 11.7488827876, 12.0533151356, 12.2205782583, 13.0649631898, 13.4694045228, 13.7650490612, 14.3456170058, 14.6786987313, 15.0854116851