| L(s) = 1 | + 5-s − 6·9-s − 8·11-s − 8·19-s + 25-s − 4·29-s + 16·31-s − 12·41-s − 6·45-s + 2·49-s − 8·55-s + 8·59-s − 4·61-s + 27·81-s − 12·89-s − 8·95-s + 48·99-s + 12·101-s + 28·109-s + 26·121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2·9-s − 2.41·11-s − 1.83·19-s + 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 0.894·45-s + 2/7·49-s − 1.07·55-s + 1.04·59-s − 0.512·61-s + 3·81-s − 1.27·89-s − 0.820·95-s + 4.82·99-s + 1.19·101-s + 2.68·109-s + 2.36·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.12246838996141367241613997715, −10.01394722702471883134476364281, −8.913111768905373027356125186253, −8.521590142691914902934262227672, −8.231140090828504156743670246342, −7.75327442992794184642667314086, −6.84926226466935514916165248985, −6.12831872993368393623278951487, −5.84906736542976130410760055690, −5.04961904822752692465387383779, −4.76905661119552301505559653794, −3.48771433097326841907897590174, −2.59137030289479413590800218969, −2.39961978181641342090490014624, 0,
2.39961978181641342090490014624, 2.59137030289479413590800218969, 3.48771433097326841907897590174, 4.76905661119552301505559653794, 5.04961904822752692465387383779, 5.84906736542976130410760055690, 6.12831872993368393623278951487, 6.84926226466935514916165248985, 7.75327442992794184642667314086, 8.231140090828504156743670246342, 8.521590142691914902934262227672, 8.913111768905373027356125186253, 10.01394722702471883134476364281, 10.12246838996141367241613997715
