| L(s) = 1 | − 2-s − 4-s − 4·5-s + 2·7-s + 8-s + 4·10-s + 2·11-s + 13-s − 2·14-s − 16-s + 3·17-s − 19-s + 4·20-s − 2·22-s − 9·23-s + 6·25-s − 26-s − 2·28-s + 29-s − 31-s + 5·32-s − 3·34-s − 8·35-s + 4·37-s + 38-s − 4·40-s + 41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 0.755·7-s + 0.353·8-s + 1.26·10-s + 0.603·11-s + 0.277·13-s − 0.534·14-s − 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.894·20-s − 0.426·22-s − 1.87·23-s + 6/5·25-s − 0.196·26-s − 0.377·28-s + 0.185·29-s − 0.179·31-s + 0.883·32-s − 0.514·34-s − 1.35·35-s + 0.657·37-s + 0.162·38-s − 0.632·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 523 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 523 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2481990412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2481990412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 523 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 24 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 15 T + 180 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 296 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−19.8480492588, −19.3953013836, −18.8418964889, −18.3366997805, −17.7461129812, −17.2515984068, −16.3397807285, −15.9452532519, −15.3076976568, −14.4980138680, −14.1516107075, −13.1540922017, −12.1397609320, −11.8210211064, −11.1960473612, −10.2935673717, −9.39546479353, −8.59170666414, −7.97590915900, −7.53724857292, −6.12345025198, −4.56274501599, −3.83200490170,
3.83200490170, 4.56274501599, 6.12345025198, 7.53724857292, 7.97590915900, 8.59170666414, 9.39546479353, 10.2935673717, 11.1960473612, 11.8210211064, 12.1397609320, 13.1540922017, 14.1516107075, 14.4980138680, 15.3076976568, 15.9452532519, 16.3397807285, 17.2515984068, 17.7461129812, 18.3366997805, 18.8418964889, 19.3953013836, 19.8480492588
