| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 3·7-s − 3·8-s − 2·9-s + 3·10-s + 11-s − 12-s − 13-s + 3·14-s + 3·15-s + 16-s + 5·17-s + 2·18-s − 8·19-s − 3·20-s + 3·21-s − 22-s + 7·23-s + 3·24-s + 26-s + 2·27-s − 3·28-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.948·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 0.774·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s − 1.83·19-s − 0.670·20-s + 0.654·21-s − 0.213·22-s + 1.45·23-s + 0.612·24-s + 0.196·26-s + 0.384·27-s − 0.566·28-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4741 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4741 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 431 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 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 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 49 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 9 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 126 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 146 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 167 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.6951254998, −17.1862084004, −16.8719786028, −16.4648147020, −15.8201027483, −15.3873870862, −14.9700760953, −14.5163820558, −13.5642471010, −12.9083220658, −12.2331414014, −12.0307530708, −11.4397378940, −10.9515256778, −10.2858225446, −9.58033526330, −9.04324842282, −8.32688303167, −7.90792151761, −6.87256183651, −6.50383116292, −5.81326735153, −4.72176268224, −3.60046348941, −2.87983906883, 0,
2.87983906883, 3.60046348941, 4.72176268224, 5.81326735153, 6.50383116292, 6.87256183651, 7.90792151761, 8.32688303167, 9.04324842282, 9.58033526330, 10.2858225446, 10.9515256778, 11.4397378940, 12.0307530708, 12.2331414014, 12.9083220658, 13.5642471010, 14.5163820558, 14.9700760953, 15.3873870862, 15.8201027483, 16.4648147020, 16.8719786028, 17.1862084004, 17.6951254998
