| L(s) = 1 | + 2·3-s − 4·4-s + 6·5-s + 3·9-s − 3·11-s − 8·12-s + 12·15-s + 12·16-s − 24·20-s + 18·23-s + 17·25-s + 4·27-s + 4·31-s − 6·33-s − 12·36-s − 8·37-s + 12·44-s + 18·45-s − 12·47-s + 24·48-s + 2·49-s − 12·53-s − 18·55-s + 12·59-s − 48·60-s − 32·64-s − 8·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 2.68·5-s + 9-s − 0.904·11-s − 2.30·12-s + 3.09·15-s + 3·16-s − 5.36·20-s + 3.75·23-s + 17/5·25-s + 0.769·27-s + 0.718·31-s − 1.04·33-s − 2·36-s − 1.31·37-s + 1.80·44-s + 2.68·45-s − 1.75·47-s + 3.46·48-s + 2/7·49-s − 1.64·53-s − 2.42·55-s + 1.56·59-s − 6.19·60-s − 4·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 314721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 314721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.038180991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.038180991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.873138966846700744402216412888, −8.618966621189813626434174913816, −8.123063877468170217097057069948, −7.60195787768679259689608146982, −6.66463575826783700250676263362, −6.65141393178900011874126957740, −5.62535288746366565403300368914, −5.18754379686639968375671320234, −5.14306131647545403565477997005, −4.59411129964790589621534187851, −3.66589986463958216194059753670, −3.00632648437536992753290178800, −2.69595897502735310420716823107, −1.70457407581975008871008195419, −1.09494185617481664600047301525,
1.09494185617481664600047301525, 1.70457407581975008871008195419, 2.69595897502735310420716823107, 3.00632648437536992753290178800, 3.66589986463958216194059753670, 4.59411129964790589621534187851, 5.14306131647545403565477997005, 5.18754379686639968375671320234, 5.62535288746366565403300368914, 6.65141393178900011874126957740, 6.66463575826783700250676263362, 7.60195787768679259689608146982, 8.123063877468170217097057069948, 8.618966621189813626434174913816, 8.873138966846700744402216412888
