| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 2·5-s + 9·6-s − 3·7-s − 3·8-s + 4·9-s + 6·10-s − 11-s − 12·12-s + 2·13-s + 9·14-s + 6·15-s + 3·16-s − 12·18-s − 19-s − 8·20-s + 9·21-s + 3·22-s − 7·23-s + 9·24-s − 25-s − 6·26-s − 6·27-s − 12·28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.894·5-s + 3.67·6-s − 1.13·7-s − 1.06·8-s + 4/3·9-s + 1.89·10-s − 0.301·11-s − 3.46·12-s + 0.554·13-s + 2.40·14-s + 1.54·15-s + 3/4·16-s − 2.82·18-s − 0.229·19-s − 1.78·20-s + 1.96·21-s + 0.639·22-s − 1.45·23-s + 1.83·24-s − 1/5·25-s − 1.17·26-s − 1.15·27-s − 2.26·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 743 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 743 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 743 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T - 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 76 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 118 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T - 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 134 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−19.7143475427, −19.5061126877, −18.8634136420, −18.4776787737, −17.7911312680, −17.6388791279, −16.9042494829, −16.3106700246, −16.1618337957, −15.5878967295, −14.6606903532, −13.4211430263, −12.6296942421, −12.0934609083, −11.3598900890, −10.8563032782, −10.2643355038, −9.52638330943, −8.99255140550, −7.88510489674, −7.47176030299, −6.23098270330, −5.75271702955, −3.96042632309, 0,
3.96042632309, 5.75271702955, 6.23098270330, 7.47176030299, 7.88510489674, 8.99255140550, 9.52638330943, 10.2643355038, 10.8563032782, 11.3598900890, 12.0934609083, 12.6296942421, 13.4211430263, 14.6606903532, 15.5878967295, 16.1618337957, 16.3106700246, 16.9042494829, 17.6388791279, 17.7911312680, 18.4776787737, 18.8634136420, 19.5061126877, 19.7143475427
