| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s + 7-s − 3·8-s − 2·9-s + 4·10-s − 11-s − 12-s − 3·13-s − 14-s + 4·15-s + 16-s − 5·17-s + 2·18-s + 7·19-s − 4·20-s − 21-s + 22-s − 2·23-s + 3·24-s + 6·25-s + 3·26-s + 2·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s + 1.26·10-s − 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 1.21·17-s + 0.471·18-s + 1.60·19-s − 0.894·20-s − 0.218·21-s + 0.213·22-s − 0.417·23-s + 0.612·24-s + 6/5·25-s + 0.588·26-s + 0.384·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5363 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5363 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 173 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 15 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 135 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−17.6906374070, −17.3583392669, −16.6349654792, −15.9547530495, −15.7506946094, −15.4656032027, −14.7378889446, −14.2497964525, −13.5684646465, −12.6741022376, −12.0429160132, −11.7465500183, −11.3980847303, −11.0383821537, −10.1262844265, −9.51843930796, −8.77958762862, −8.12076079353, −7.88518360330, −6.94970428794, −6.52024082962, −5.36262957837, −4.76925862206, −3.66145587473, −2.68415271306, 0,
2.68415271306, 3.66145587473, 4.76925862206, 5.36262957837, 6.52024082962, 6.94970428794, 7.88518360330, 8.12076079353, 8.77958762862, 9.51843930796, 10.1262844265, 11.0383821537, 11.3980847303, 11.7465500183, 12.0429160132, 12.6741022376, 13.5684646465, 14.2497964525, 14.7378889446, 15.4656032027, 15.7506946094, 15.9547530495, 16.6349654792, 17.3583392669, 17.6906374070
