L(s) = 1 | − 4·5-s − 2·9-s + 2·11-s + 13-s − 2·17-s + 2·19-s + 3·23-s + 6·25-s + 3·27-s − 29-s + 3·31-s − 12·37-s + 41-s − 4·43-s + 8·45-s + 3·47-s + 10·49-s − 4·53-s − 8·55-s − 8·59-s − 8·61-s − 4·65-s + 2·67-s + 7·71-s + 11·73-s − 10·79-s + 4·81-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 0.625·23-s + 6/5·25-s + 0.577·27-s − 0.185·29-s + 0.538·31-s − 1.97·37-s + 0.156·41-s − 0.609·43-s + 1.19·45-s + 0.437·47-s + 10/7·49-s − 0.549·53-s − 1.07·55-s − 1.04·59-s − 1.02·61-s − 0.496·65-s + 0.244·67-s + 0.830·71-s + 1.28·73-s − 1.12·79-s + 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4453748542\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4453748542\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 11 T + 124 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.8278713562, −19.4149334317, −18.8035573521, −18.3822107134, −17.4425043230, −17.0875875799, −16.3828873887, −15.7665595893, −15.3664104464, −14.9256317224, −13.9745060942, −13.7262317709, −12.5219615781, −12.1574493503, −11.5467710813, −11.0772530440, −10.3856958944, −9.19706065176, −8.66273573031, −7.97621904415, −7.22942070850, −6.45160845933, −5.18110661744, −4.13114025718, −3.25020887365,
3.25020887365, 4.13114025718, 5.18110661744, 6.45160845933, 7.22942070850, 7.97621904415, 8.66273573031, 9.19706065176, 10.3856958944, 11.0772530440, 11.5467710813, 12.1574493503, 12.5219615781, 13.7262317709, 13.9745060942, 14.9256317224, 15.3664104464, 15.7665595893, 16.3828873887, 17.0875875799, 17.4425043230, 18.3822107134, 18.8035573521, 19.4149334317, 19.8278713562