| L(s) = 1 | + 4-s + 8·5-s − 6·9-s + 2·11-s + 16-s + 8·20-s + 2·23-s + 38·25-s − 6·36-s − 8·37-s + 2·44-s − 48·45-s + 2·49-s − 8·53-s + 16·55-s + 24·59-s + 64-s − 20·67-s + 8·80-s + 27·81-s − 12·89-s + 2·92-s + 12·97-s − 12·99-s + 38·100-s − 16·103-s − 28·113-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.57·5-s − 2·9-s + 0.603·11-s + 1/4·16-s + 1.78·20-s + 0.417·23-s + 38/5·25-s − 36-s − 1.31·37-s + 0.301·44-s − 7.15·45-s + 2/7·49-s − 1.09·53-s + 2.15·55-s + 3.12·59-s + 1/8·64-s − 2.44·67-s + 0.894·80-s + 3·81-s − 1.27·89-s + 0.208·92-s + 1.21·97-s − 1.20·99-s + 19/5·100-s − 1.57·103-s − 2.63·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.623550713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.623550713\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.971935452602447588346290189052, −8.658761946253597060620751515153, −8.318494750941895003110401949320, −7.29335654603312257574422641336, −6.69441913641732873591204403077, −6.47661544722030437177591118449, −5.84432417959727908710252168629, −5.77239746630691660444468744425, −5.25822720676538701581916863445, −4.89394645249622774418243112804, −3.61702520706098311996149342232, −2.73508823518418390595732765608, −2.63232737474552247848840602025, −1.89613910278618470304457940228, −1.29994774368489812843054385244,
1.29994774368489812843054385244, 1.89613910278618470304457940228, 2.63232737474552247848840602025, 2.73508823518418390595732765608, 3.61702520706098311996149342232, 4.89394645249622774418243112804, 5.25822720676538701581916863445, 5.77239746630691660444468744425, 5.84432417959727908710252168629, 6.47661544722030437177591118449, 6.69441913641732873591204403077, 7.29335654603312257574422641336, 8.318494750941895003110401949320, 8.658761946253597060620751515153, 8.971935452602447588346290189052
