| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 2·9-s − 2·11-s − 12-s − 3·13-s + 14-s + 16-s − 17-s + 2·18-s − 2·19-s − 21-s − 2·22-s + 4·23-s − 24-s − 3·26-s − 6·27-s + 28-s − 4·31-s + 32-s + 2·33-s − 34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.603·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 0.458·19-s − 0.218·21-s − 0.426·22-s + 0.834·23-s − 0.204·24-s − 0.588·26-s − 1.15·27-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.096044510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096044510\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 373 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 51 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 72 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 81 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−17.0448564572, −16.7995590236, −16.3248428050, −15.5559151892, −15.2659954989, −14.7534632860, −14.3197619278, −13.4468223988, −13.1482640541, −12.6478634313, −12.1472528138, −11.3232107171, −11.2296908388, −10.4317893595, −9.90950455121, −9.26205453070, −8.35731098558, −7.62661790269, −7.15617090436, −6.38807263006, −5.65741600453, −4.89951769402, −4.45926429523, −3.28572468680, −2.04432735959,
2.04432735959, 3.28572468680, 4.45926429523, 4.89951769402, 5.65741600453, 6.38807263006, 7.15617090436, 7.62661790269, 8.35731098558, 9.26205453070, 9.90950455121, 10.4317893595, 11.2296908388, 11.3232107171, 12.1472528138, 12.6478634313, 13.1482640541, 13.4468223988, 14.3197619278, 14.7534632860, 15.2659954989, 15.5559151892, 16.3248428050, 16.7995590236, 17.0448564572
