| L(s) = 1 | + 2-s + 4-s + 8-s + 16-s − 12·17-s + 4·19-s − 10·25-s + 32-s − 12·34-s + 4·38-s − 12·41-s + 16·43-s + 49-s − 10·50-s + 12·59-s + 64-s − 8·67-s − 12·68-s + 4·73-s + 4·76-s − 12·82-s + 12·83-s + 16·86-s + 12·89-s − 20·97-s + 98-s − 10·100-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s − 2.91·17-s + 0.917·19-s − 2·25-s + 0.176·32-s − 2.05·34-s + 0.648·38-s − 1.87·41-s + 2.43·43-s + 1/7·49-s − 1.41·50-s + 1.56·59-s + 1/8·64-s − 0.977·67-s − 1.45·68-s + 0.468·73-s + 0.458·76-s − 1.32·82-s + 1.31·83-s + 1.72·86-s + 1.27·89-s − 2.03·97-s + 0.101·98-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.146186617845743660921903375935, −7.80365099617056766338064266882, −7.31446556680916748478810640735, −6.71310127772875365703449938682, −6.52736531410894424318591996955, −5.98578739746113342675162950857, −5.31921059379060528260176389250, −5.10494312459275299769631650208, −4.19102102405251164216473944262, −4.13413144386183145123967904062, −3.51719354828858627431019020699, −2.45963109762455912567028940336, −2.39101862347891857041372548490, −1.45076284792606367019758683143, 0,
1.45076284792606367019758683143, 2.39101862347891857041372548490, 2.45963109762455912567028940336, 3.51719354828858627431019020699, 4.13413144386183145123967904062, 4.19102102405251164216473944262, 5.10494312459275299769631650208, 5.31921059379060528260176389250, 5.98578739746113342675162950857, 6.52736531410894424318591996955, 6.71310127772875365703449938682, 7.31446556680916748478810640735, 7.80365099617056766338064266882, 8.146186617845743660921903375935
