L(s) = 1 | + 2·9-s + 6·11-s − 4·19-s − 18·29-s + 16·31-s − 12·41-s − 49-s − 20·61-s + 30·71-s + 14·79-s − 5·81-s + 24·89-s + 12·99-s + 36·101-s + 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + ⋯ |
L(s) = 1 | + 2/3·9-s + 1.80·11-s − 0.917·19-s − 3.34·29-s + 2.87·31-s − 1.87·41-s − 1/7·49-s − 2.56·61-s + 3.56·71-s + 1.57·79-s − 5/9·81-s + 2.54·89-s + 1.20·99-s + 3.58·101-s + 1.34·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.137763132\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.137763132\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.65137952057750505305899590694, −10.22878760541338227363484868375, −9.701215662760807156810281907471, −9.250210507348614390827957263560, −9.244706208696706381131837475132, −8.437478049592419141278881755028, −8.179829966329081059073020805028, −7.57488352665804640850623102093, −7.12682876507005590906700878798, −6.64545750706687171589924825427, −6.24198449306097599567071623280, −6.00621160863797576205116566336, −5.10995238471114534684013773628, −4.65311937294734594598776842469, −4.23410994602927263536481362554, −3.46289100504918433249554152372, −3.44655202647465720328780809249, −2.05982412550434002318841970182, −1.83553176259099801294864750371, −0.805558116170743570308512986130,
0.805558116170743570308512986130, 1.83553176259099801294864750371, 2.05982412550434002318841970182, 3.44655202647465720328780809249, 3.46289100504918433249554152372, 4.23410994602927263536481362554, 4.65311937294734594598776842469, 5.10995238471114534684013773628, 6.00621160863797576205116566336, 6.24198449306097599567071623280, 6.64545750706687171589924825427, 7.12682876507005590906700878798, 7.57488352665804640850623102093, 8.179829966329081059073020805028, 8.437478049592419141278881755028, 9.244706208696706381131837475132, 9.250210507348614390827957263560, 9.701215662760807156810281907471, 10.22878760541338227363484868375, 10.65137952057750505305899590694