| L(s) = 1 | + 2-s + 4-s + 8-s + 16-s − 6·19-s − 4·25-s − 4·29-s + 32-s − 6·38-s − 16·41-s − 10·49-s − 4·50-s + 8·53-s − 4·58-s − 4·59-s − 12·61-s + 64-s + 4·71-s + 4·73-s − 6·76-s − 16·82-s + 12·89-s − 10·98-s − 4·100-s + 8·106-s − 20·107-s − 4·116-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s − 1.37·19-s − 4/5·25-s − 0.742·29-s + 0.176·32-s − 0.973·38-s − 2.49·41-s − 1.42·49-s − 0.565·50-s + 1.09·53-s − 0.525·58-s − 0.520·59-s − 1.53·61-s + 1/8·64-s + 0.474·71-s + 0.468·73-s − 0.688·76-s − 1.76·82-s + 1.27·89-s − 1.01·98-s − 2/5·100-s + 0.777·106-s − 1.93·107-s − 0.371·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−8.265984820090898448093352791431, −7.904100493342645029718157344745, −7.33937922299687593527494893639, −6.82991812998689751532316502026, −6.45125757980468628828621814374, −6.01214106883646312666091813481, −5.51398885957276924694108318079, −4.88211461497608772683371286276, −4.61626945576028703283761741409, −3.79607823363534491711807649039, −3.58598500109909191279134422362, −2.80084438245048525437172844010, −2.07286600693385220244828115205, −1.56128284079780541834205616601, 0,
1.56128284079780541834205616601, 2.07286600693385220244828115205, 2.80084438245048525437172844010, 3.58598500109909191279134422362, 3.79607823363534491711807649039, 4.61626945576028703283761741409, 4.88211461497608772683371286276, 5.51398885957276924694108318079, 6.01214106883646312666091813481, 6.45125757980468628828621814374, 6.82991812998689751532316502026, 7.33937922299687593527494893639, 7.904100493342645029718157344745, 8.265984820090898448093352791431
