| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 5·9-s + 2·14-s + 16-s + 6·17-s + 5·18-s + 6·23-s − 10·25-s − 2·28-s − 8·31-s − 32-s − 6·34-s − 5·36-s − 6·46-s − 11·49-s + 10·50-s + 2·56-s + 8·62-s + 10·63-s + 64-s + 6·68-s − 12·71-s + 5·72-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 5/3·9-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 1.17·18-s + 1.25·23-s − 2·25-s − 0.377·28-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 5/6·36-s − 0.884·46-s − 1.57·49-s + 1.41·50-s + 0.267·56-s + 1.01·62-s + 1.25·63-s + 1/8·64-s + 0.727·68-s − 1.42·71-s + 0.589·72-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.972461666120437498884029821538, −9.430534638052368973397733045259, −8.684413720393242419277779815238, −8.666881218945426773456780708719, −7.73591843742988285704805588989, −7.49822591496797438401690336538, −6.79687210361750407728091466084, −5.93366234401884381484558908154, −5.81027585652647529822118508915, −5.20686401818851827647471141246, −4.07138629192131545636061302538, −3.12424893607844884324162519175, −2.99432139105097413830284496966, −1.68171654176783246558951167350, 0,
1.68171654176783246558951167350, 2.99432139105097413830284496966, 3.12424893607844884324162519175, 4.07138629192131545636061302538, 5.20686401818851827647471141246, 5.81027585652647529822118508915, 5.93366234401884381484558908154, 6.79687210361750407728091466084, 7.49822591496797438401690336538, 7.73591843742988285704805588989, 8.666881218945426773456780708719, 8.684413720393242419277779815238, 9.430534638052368973397733045259, 9.972461666120437498884029821538
