L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·5-s + 2·6-s − 4·7-s − 3·8-s + 3·9-s + 2·10-s + 4·11-s − 4·12-s − 2·13-s − 4·14-s + 4·15-s + 16-s + 6·17-s + 3·18-s − 10·19-s − 4·20-s − 8·21-s + 4·22-s − 2·23-s − 6·24-s − 2·25-s − 2·26-s + 4·27-s + 8·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.894·5-s + 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s + 0.632·10-s + 1.20·11-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 2.29·19-s − 0.894·20-s − 1.74·21-s + 0.852·22-s − 0.417·23-s − 1.22·24-s − 2/5·25-s − 0.392·26-s + 0.769·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.432343627\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.432343627\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 30 T + 378 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T - 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.19656138291085195577105289556, −14.13784562908028887463274991736, −13.30822535943976405292619280405, −13.08126410111092776195400260092, −12.54842031959081348843701926982, −12.38690658068640418137094254295, −11.31442079433776982465479980330, −10.25919111520854853113468939749, −9.848199667372640298808001683822, −9.594911363765538106327790540763, −8.810338225621425421133394361693, −8.787088278780598681714873590880, −7.64544354559500736570238313155, −6.96227678862324707950148129167, −5.92447139966814564047583105421, −5.91703822599761461769955743526, −4.48583633468729403714404321989, −3.96740643999368951175333635084, −3.33284344106940796359913592368, −2.20009654735353763797367045973,
2.20009654735353763797367045973, 3.33284344106940796359913592368, 3.96740643999368951175333635084, 4.48583633468729403714404321989, 5.91703822599761461769955743526, 5.92447139966814564047583105421, 6.96227678862324707950148129167, 7.64544354559500736570238313155, 8.787088278780598681714873590880, 8.810338225621425421133394361693, 9.594911363765538106327790540763, 9.848199667372640298808001683822, 10.25919111520854853113468939749, 11.31442079433776982465479980330, 12.38690658068640418137094254295, 12.54842031959081348843701926982, 13.08126410111092776195400260092, 13.30822535943976405292619280405, 14.13784562908028887463274991736, 14.19656138291085195577105289556