| L(s) = 1 | − 2-s − 4-s − 5-s + 5·7-s + 8-s + 9-s + 10-s − 11-s − 6·13-s − 5·14-s − 16-s − 17-s − 18-s + 19-s + 20-s + 22-s − 9·23-s − 2·25-s + 6·26-s − 5·28-s + 3·29-s − 6·31-s + 5·32-s + 34-s − 5·35-s − 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 1.66·13-s − 1.33·14-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.213·22-s − 1.87·23-s − 2/5·25-s + 1.17·26-s − 0.944·28-s + 0.557·29-s − 1.07·31-s + 0.883·32-s + 0.171·34-s − 0.845·35-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56061 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56061 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 6229 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_4$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17 T + 147 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 113 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 186 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−14.8861950443, −14.2811349364, −14.1339729146, −13.6644301304, −13.0650407122, −12.3749591312, −12.0335837922, −11.7575383522, −11.1235322694, −10.7478096550, −10.0824300425, −9.74771546742, −9.26789912878, −8.68823205254, −8.06883775758, −7.82460595198, −7.60410109638, −6.76301572370, −6.04335617149, −5.10561124995, −4.82425859868, −4.41554090174, −3.56810110969, −2.34368318708, −1.68325928564, 0,
1.68325928564, 2.34368318708, 3.56810110969, 4.41554090174, 4.82425859868, 5.10561124995, 6.04335617149, 6.76301572370, 7.60410109638, 7.82460595198, 8.06883775758, 8.68823205254, 9.26789912878, 9.74771546742, 10.0824300425, 10.7478096550, 11.1235322694, 11.7575383522, 12.0335837922, 12.3749591312, 13.0650407122, 13.6644301304, 14.1339729146, 14.2811349364, 14.8861950443
