| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·9-s + 11-s + 14-s + 16-s + 4·18-s + 3·19-s − 22-s − 23-s − 2·25-s − 28-s + 5·29-s + 31-s − 32-s − 4·36-s − 8·37-s − 3·38-s + 2·41-s − 2·43-s + 44-s + 46-s − 47-s − 5·49-s + 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 4/3·9-s + 0.301·11-s + 0.267·14-s + 1/4·16-s + 0.942·18-s + 0.688·19-s − 0.213·22-s − 0.208·23-s − 2/5·25-s − 0.188·28-s + 0.928·29-s + 0.179·31-s − 0.176·32-s − 2/3·36-s − 1.31·37-s − 0.486·38-s + 0.312·41-s − 0.304·43-s + 0.150·44-s + 0.147·46-s − 0.145·47-s − 5/7·49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 146992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 146992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 9187 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 56 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + T + 50 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 106 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 192 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 17 T + 202 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 76 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−14.0749198161, −13.5914916520, −12.9447926144, −12.5913019941, −12.0449976034, −11.6217244071, −11.3966107834, −10.9496449257, −10.2429487647, −10.0050869537, −9.55657154178, −8.92005803319, −8.66916021355, −8.18579757991, −7.76831619969, −6.99263375824, −6.79383198283, −5.94558766562, −5.80882703888, −5.07105713787, −4.39966243550, −3.46952078382, −3.08372024999, −2.37388407512, −1.36179276609, 0,
1.36179276609, 2.37388407512, 3.08372024999, 3.46952078382, 4.39966243550, 5.07105713787, 5.80882703888, 5.94558766562, 6.79383198283, 6.99263375824, 7.76831619969, 8.18579757991, 8.66916021355, 8.92005803319, 9.55657154178, 10.0050869537, 10.2429487647, 10.9496449257, 11.3966107834, 11.6217244071, 12.0449976034, 12.5913019941, 12.9447926144, 13.5914916520, 14.0749198161
