| L(s) = 1 | + 4-s − 2·7-s − 3·9-s + 4·13-s + 16-s − 6·25-s − 2·28-s − 16·31-s − 3·36-s − 4·37-s + 8·43-s + 3·49-s + 4·52-s + 20·61-s + 6·63-s + 64-s − 24·67-s − 28·73-s + 9·81-s − 8·91-s + 20·97-s − 6·100-s − 4·109-s − 2·112-s − 12·117-s + 121-s − 16·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 9-s + 1.10·13-s + 1/4·16-s − 6/5·25-s − 0.377·28-s − 2.87·31-s − 1/2·36-s − 0.657·37-s + 1.21·43-s + 3/7·49-s + 0.554·52-s + 2.56·61-s + 0.755·63-s + 1/8·64-s − 2.93·67-s − 3.27·73-s + 81-s − 0.838·91-s + 2.03·97-s − 3/5·100-s − 0.383·109-s − 0.188·112-s − 1.10·117-s + 1/11·121-s − 1.43·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.919520401542258285479247886410, −8.582739275429656645366707671354, −7.58187479426412561326624344048, −7.52239117864584703743055005677, −6.96355267990089995055745653875, −6.08896062740208954182977845395, −6.02643591301064267650899026350, −5.60617960823028323278478834583, −4.92364198553775270952134324473, −3.79941779708894965776977767454, −3.78476385235783926950251020606, −3.00404374709518806814432634087, −2.30984363521644281586554494696, −1.49721415431484567819975457145, 0,
1.49721415431484567819975457145, 2.30984363521644281586554494696, 3.00404374709518806814432634087, 3.78476385235783926950251020606, 3.79941779708894965776977767454, 4.92364198553775270952134324473, 5.60617960823028323278478834583, 6.02643591301064267650899026350, 6.08896062740208954182977845395, 6.96355267990089995055745653875, 7.52239117864584703743055005677, 7.58187479426412561326624344048, 8.582739275429656645366707671354, 8.919520401542258285479247886410
