| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s − 6·11-s − 2·12-s + 16-s + 6·17-s − 3·18-s + 10·19-s − 6·22-s − 2·24-s + 14·27-s + 32-s + 12·33-s + 6·34-s − 3·36-s + 10·38-s − 6·41-s + 8·43-s − 6·44-s − 2·48-s − 10·49-s − 12·51-s + 14·54-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s − 1.80·11-s − 0.577·12-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 2.29·19-s − 1.27·22-s − 0.408·24-s + 2.69·27-s + 0.176·32-s + 2.08·33-s + 1.02·34-s − 1/2·36-s + 1.62·38-s − 0.937·41-s + 1.21·43-s − 0.904·44-s − 0.288·48-s − 1.42·49-s − 1.68·51-s + 1.90·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208445557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208445557\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.930723730317650982781533521356, −9.429239826768018761346169045309, −8.592914531793421630007930470901, −8.071729218562685355881418106440, −7.63971798893296337235505790724, −7.23606762344534562872845833638, −6.35306147256574072432179047412, −5.91956414482903908532570774168, −5.34945783064988058695168525193, −5.22748830252304089500003968022, −4.78507259529948280984941273759, −3.29699975126484092788987476482, −3.27747260558597777923671030481, −2.37228414250853175167766648204, −0.806837847292090646690279005746,
0.806837847292090646690279005746, 2.37228414250853175167766648204, 3.27747260558597777923671030481, 3.29699975126484092788987476482, 4.78507259529948280984941273759, 5.22748830252304089500003968022, 5.34945783064988058695168525193, 5.91956414482903908532570774168, 6.35306147256574072432179047412, 7.23606762344534562872845833638, 7.63971798893296337235505790724, 8.071729218562685355881418106440, 8.592914531793421630007930470901, 9.429239826768018761346169045309, 9.930723730317650982781533521356
