| L(s) = 1 | − 3-s + 4-s − 4·5-s − 7-s − 2·9-s + 5·11-s − 12-s − 2·13-s + 4·15-s + 16-s − 2·17-s − 5·19-s − 4·20-s + 21-s − 8·23-s + 4·25-s + 2·27-s − 28-s − 5·29-s − 2·31-s − 5·33-s + 4·35-s − 2·36-s − 37-s + 2·39-s + 9·41-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.288·12-s − 0.554·13-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 1.14·19-s − 0.894·20-s + 0.218·21-s − 1.66·23-s + 4/5·25-s + 0.384·27-s − 0.188·28-s − 0.928·29-s − 0.359·31-s − 0.870·33-s + 0.676·35-s − 1/3·36-s − 0.164·37-s + 0.320·39-s + 1.40·41-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16972 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16972 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 4243 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 107 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 26 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 86 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 128 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 159 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−16.0851909300, −15.9024124128, −15.2281381981, −14.7914110779, −14.4903569385, −13.9558926766, −13.0582599310, −12.6239796049, −12.0096799788, −11.7596530968, −11.4434682020, −11.0316756574, −10.3623393355, −9.70527767722, −8.99722850683, −8.52741592698, −7.85166102641, −7.39041737369, −6.77936727625, −6.11334708262, −5.72210626196, −4.42535695906, −4.06707527391, −3.42051366618, −2.14476907069, 0,
2.14476907069, 3.42051366618, 4.06707527391, 4.42535695906, 5.72210626196, 6.11334708262, 6.77936727625, 7.39041737369, 7.85166102641, 8.52741592698, 8.99722850683, 9.70527767722, 10.3623393355, 11.0316756574, 11.4434682020, 11.7596530968, 12.0096799788, 12.6239796049, 13.0582599310, 13.9558926766, 14.4903569385, 14.7914110779, 15.2281381981, 15.9024124128, 16.0851909300
