| L(s) = 1 | − 2·2-s + 2·4-s − 6·7-s + 12·14-s − 4·16-s − 2·23-s − 12·28-s + 8·32-s − 24·41-s − 18·43-s + 4·46-s − 14·47-s + 18·49-s − 16·61-s − 8·64-s − 6·67-s + 48·82-s − 22·83-s + 36·86-s − 4·92-s + 28·94-s − 36·98-s + 36·101-s − 18·103-s + 26·107-s + 24·112-s + 22·121-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.26·7-s + 3.20·14-s − 16-s − 0.417·23-s − 2.26·28-s + 1.41·32-s − 3.74·41-s − 2.74·43-s + 0.589·46-s − 2.04·47-s + 18/7·49-s − 2.04·61-s − 64-s − 0.733·67-s + 5.30·82-s − 2.41·83-s + 3.88·86-s − 0.417·92-s + 2.88·94-s − 3.63·98-s + 3.58·101-s − 1.77·103-s + 2.51·107-s + 2.26·112-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−9.840646564602905882016952216415, −9.783177142889602176615982615267, −8.882969361526073725695830923503, −8.874163464935605254975256118737, −8.339160821893038661546252725528, −7.980327520811191375411657917353, −7.22506277653226747992125334886, −7.04592510983331967017197185623, −6.48625854283405323859461544741, −6.36111142548848000335399366700, −5.79875495962519145827089414206, −4.86774410197011933552504603467, −4.76605875059416824221828872885, −3.69955452081662181540677535696, −3.22214226622701215711523678670, −3.09261678250413111831336110729, −1.97857098273931622214722382350, −1.49522296593912016389884812796, 0, 0,
1.49522296593912016389884812796, 1.97857098273931622214722382350, 3.09261678250413111831336110729, 3.22214226622701215711523678670, 3.69955452081662181540677535696, 4.76605875059416824221828872885, 4.86774410197011933552504603467, 5.79875495962519145827089414206, 6.36111142548848000335399366700, 6.48625854283405323859461544741, 7.04592510983331967017197185623, 7.22506277653226747992125334886, 7.980327520811191375411657917353, 8.339160821893038661546252725528, 8.874163464935605254975256118737, 8.882969361526073725695830923503, 9.783177142889602176615982615267, 9.840646564602905882016952216415
