| L(s) = 1 | − 2·3-s − 2·7-s + 2·9-s + 8·11-s + 6·13-s + 6·17-s + 4·21-s − 6·23-s − 6·27-s + 4·29-s − 16·33-s + 6·37-s − 12·39-s + 12·41-s + 6·43-s − 18·47-s + 2·49-s − 12·51-s − 10·53-s − 4·63-s + 18·67-s + 12·69-s − 10·73-s − 16·77-s + 11·81-s + 6·83-s − 8·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 2/3·9-s + 2.41·11-s + 1.66·13-s + 1.45·17-s + 0.872·21-s − 1.25·23-s − 1.15·27-s + 0.742·29-s − 2.78·33-s + 0.986·37-s − 1.92·39-s + 1.87·41-s + 0.914·43-s − 2.62·47-s + 2/7·49-s − 1.68·51-s − 1.37·53-s − 0.503·63-s + 2.19·67-s + 1.44·69-s − 1.17·73-s − 1.82·77-s + 11/9·81-s + 0.658·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801707467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801707467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.585877442308488987326388676005, −9.450063801991801703020153875300, −8.981171024730539065000692184466, −8.223414305163055493358569399233, −8.149386727088358790209677125127, −7.62448348572277372206878590224, −7.02989514194210897978227581508, −6.45207936515420238596367940894, −6.28092662299837410457452354841, −6.21990880438484251376851501867, −5.65825239570499149942260611104, −5.27624821061915138889392810623, −4.50389945906390817817446593542, −3.97986816408030996002268447173, −3.86454480606741618614138680070, −3.37360755439257778888374860730, −2.70036591644518861487848003218, −1.61465793991709838519340038760, −1.30952014260074673879855641354, −0.64783116632890172521948894174,
0.64783116632890172521948894174, 1.30952014260074673879855641354, 1.61465793991709838519340038760, 2.70036591644518861487848003218, 3.37360755439257778888374860730, 3.86454480606741618614138680070, 3.97986816408030996002268447173, 4.50389945906390817817446593542, 5.27624821061915138889392810623, 5.65825239570499149942260611104, 6.21990880438484251376851501867, 6.28092662299837410457452354841, 6.45207936515420238596367940894, 7.02989514194210897978227581508, 7.62448348572277372206878590224, 8.149386727088358790209677125127, 8.223414305163055493358569399233, 8.981171024730539065000692184466, 9.450063801991801703020153875300, 9.585877442308488987326388676005
