| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 2·5-s + 6·6-s − 2·7-s + 3·9-s + 4·10-s + 11-s − 3·12-s + 4·14-s + 6·15-s + 16-s − 5·17-s − 6·18-s + 3·19-s − 2·20-s + 6·21-s − 2·22-s + 4·23-s + 2·25-s − 2·28-s − 16·29-s − 12·30-s + 2·32-s − 3·33-s + 10·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s + 2.44·6-s − 0.755·7-s + 9-s + 1.26·10-s + 0.301·11-s − 0.866·12-s + 1.06·14-s + 1.54·15-s + 1/4·16-s − 1.21·17-s − 1.41·18-s + 0.688·19-s − 0.447·20-s + 1.30·21-s − 0.426·22-s + 0.834·23-s + 2/5·25-s − 0.377·28-s − 2.97·29-s − 2.19·30-s + 0.353·32-s − 0.522·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 229 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 3 T - 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 88 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−19.8522384327, −19.1366480818, −18.8660353885, −18.2713346022, −17.8393783226, −17.2047877083, −16.9250209594, −16.4875981644, −15.8471334463, −15.2668029432, −14.6771284304, −13.3311131854, −13.0607834347, −11.9984811963, −11.6637046398, −11.1577942373, −10.5492761536, −9.72147379610, −9.09730519925, −8.54635128717, −7.43506310856, −6.83944361340, −5.92006913258, −5.08503668609, −3.70645262249, 0,
3.70645262249, 5.08503668609, 5.92006913258, 6.83944361340, 7.43506310856, 8.54635128717, 9.09730519925, 9.72147379610, 10.5492761536, 11.1577942373, 11.6637046398, 11.9984811963, 13.0607834347, 13.3311131854, 14.6771284304, 15.2668029432, 15.8471334463, 16.4875981644, 16.9250209594, 17.2047877083, 17.8393783226, 18.2713346022, 18.8660353885, 19.1366480818, 19.8522384327
