| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9854263883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9854263883\) |
| \(L(\frac{3}{2})\) |
|
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
−11.52426214006865205005559601963, −11.45666669128344280655084086532, −10.83635878381861701657899519690, −9.851840183892789826086741161768, −9.267253735733507250496452791400, −9.022035638415746178025932660186, −8.516348691113562212594470236153, −7.50932176073420730404294525891, −6.97238544391138654785050907556, −6.11165748014924370206663675456, −5.70093888866808737514223874840, −5.01978566915858300867155671122, −3.66795927025167344126272800405, −3.26180587408034592610487247010, −1.73185811467864323225134521099,
1.73185811467864323225134521099, 3.26180587408034592610487247010, 3.66795927025167344126272800405, 5.01978566915858300867155671122, 5.70093888866808737514223874840, 6.11165748014924370206663675456, 6.97238544391138654785050907556, 7.50932176073420730404294525891, 8.516348691113562212594470236153, 9.022035638415746178025932660186, 9.267253735733507250496452791400, 9.851840183892789826086741161768, 10.83635878381861701657899519690, 11.45666669128344280655084086532, 11.52426214006865205005559601963
