| L(s) = 1 | + 2·3-s − 8·5-s + 7-s + 9-s − 16·15-s − 4·17-s + 2·21-s + 38·25-s − 4·27-s − 8·35-s − 12·37-s − 4·41-s + 16·43-s − 8·45-s − 8·47-s + 49-s − 8·51-s + 12·59-s + 63-s − 24·67-s + 76·75-s − 16·79-s − 11·81-s + 12·83-s + 32·85-s + 20·89-s + 24·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3.57·5-s + 0.377·7-s + 1/3·9-s − 4.13·15-s − 0.970·17-s + 0.436·21-s + 38/5·25-s − 0.769·27-s − 1.35·35-s − 1.97·37-s − 0.624·41-s + 2.43·43-s − 1.19·45-s − 1.16·47-s + 1/7·49-s − 1.12·51-s + 1.56·59-s + 0.125·63-s − 2.93·67-s + 8.77·75-s − 1.80·79-s − 1.22·81-s + 1.31·83-s + 3.47·85-s + 2.11·89-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7844847992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7844847992\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.928329697232844766585907221767, −8.515074793505794327617365568851, −8.188057214509100026281306554311, −7.69639740408856738387093436756, −7.36849069017085576948583776260, −7.14438434722230698367292292955, −6.46556088998900741811578342958, −5.49102837407369040609817160877, −4.54986168178647908951568994162, −4.50185552371175935345657331179, −3.89113776702465302721269609787, −3.28822043116364534811243543739, −3.14137792992390816749783057891, −2.03505353231784433825131603728, −0.51874703726271079374177335007,
0.51874703726271079374177335007, 2.03505353231784433825131603728, 3.14137792992390816749783057891, 3.28822043116364534811243543739, 3.89113776702465302721269609787, 4.50185552371175935345657331179, 4.54986168178647908951568994162, 5.49102837407369040609817160877, 6.46556088998900741811578342958, 7.14438434722230698367292292955, 7.36849069017085576948583776260, 7.69639740408856738387093436756, 8.188057214509100026281306554311, 8.515074793505794327617365568851, 8.928329697232844766585907221767
