| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s + 8-s + 10-s + 2·11-s − 2·13-s + 14-s − 3·16-s + 4·17-s − 9·19-s − 20-s − 2·22-s − 9·23-s − 2·25-s + 2·26-s − 28-s − 3·29-s + 10·31-s + 5·32-s − 4·34-s + 35-s + 5·37-s + 9·38-s − 40-s + 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.554·13-s + 0.267·14-s − 3/4·16-s + 0.970·17-s − 2.06·19-s − 0.223·20-s − 0.426·22-s − 1.87·23-s − 2/5·25-s + 0.392·26-s − 0.188·28-s − 0.557·29-s + 1.79·31-s + 0.883·32-s − 0.685·34-s + 0.169·35-s + 0.821·37-s + 1.45·38-s − 0.158·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153022 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153022 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 76511 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 363 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 84 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 129 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.0936965267, −13.4501195583, −12.8660077296, −12.4833426373, −12.1989044348, −11.6471582770, −11.2883391006, −10.7503599173, −10.4125303448, −9.81035847586, −9.62177798772, −9.14117822241, −8.34922199480, −8.04753510749, −7.81213010752, −7.19045534712, −6.53067271654, −6.21501472883, −5.78284846139, −4.74935965157, −4.20251372219, −3.98958139659, −2.88735277120, −2.27517067208, −1.40846366702, 0,
1.40846366702, 2.27517067208, 2.88735277120, 3.98958139659, 4.20251372219, 4.74935965157, 5.78284846139, 6.21501472883, 6.53067271654, 7.19045534712, 7.81213010752, 8.04753510749, 8.34922199480, 9.14117822241, 9.62177798772, 9.81035847586, 10.4125303448, 10.7503599173, 11.2883391006, 11.6471582770, 12.1989044348, 12.4833426373, 12.8660077296, 13.4501195583, 14.0936965267
