| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s + 5·7-s + 8-s + 4·9-s − 10-s − 2·12-s + 5·13-s − 5·14-s − 2·15-s − 3·16-s − 7·17-s − 4·18-s + 3·19-s + 20-s − 10·21-s − 6·23-s − 2·24-s − 7·25-s − 5·26-s − 5·27-s + 5·28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 1.88·7-s + 0.353·8-s + 4/3·9-s − 0.316·10-s − 0.577·12-s + 1.38·13-s − 1.33·14-s − 0.516·15-s − 3/4·16-s − 1.69·17-s − 0.942·18-s + 0.688·19-s + 0.223·20-s − 2.18·21-s − 1.25·23-s − 0.408·24-s − 7/5·25-s − 0.980·26-s − 0.962·27-s + 0.944·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15126 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15126 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7840868078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7840868078\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 2521 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 19 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 25 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 72 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 133 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 128 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.1512543213, −15.9469616005, −15.3521048119, −14.6302579539, −14.1468395022, −13.5120427928, −13.3363218177, −12.5724832146, −11.7129025404, −11.4590525191, −11.2085837398, −10.6890470776, −10.1929935295, −9.63337010088, −8.85812470055, −8.26003529254, −7.87643980012, −7.15485944272, −6.51713376412, −5.93861981711, −5.23795901954, −4.51784494396, −4.03879840927, −2.03651255680, −1.46849842500,
1.46849842500, 2.03651255680, 4.03879840927, 4.51784494396, 5.23795901954, 5.93861981711, 6.51713376412, 7.15485944272, 7.87643980012, 8.26003529254, 8.85812470055, 9.63337010088, 10.1929935295, 10.6890470776, 11.2085837398, 11.4590525191, 11.7129025404, 12.5724832146, 13.3363218177, 13.5120427928, 14.1468395022, 14.6302579539, 15.3521048119, 15.9469616005, 16.1512543213
