| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 6·11-s + 12-s − 8·13-s + 16-s + 2·18-s + 6·22-s + 12·23-s − 24-s + 8·26-s − 5·27-s − 32-s − 6·33-s − 2·36-s + 4·37-s − 8·39-s − 6·44-s − 12·46-s + 24·47-s + 48-s − 10·49-s − 8·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 0.471·18-s + 1.27·22-s + 2.50·23-s − 0.204·24-s + 1.56·26-s − 0.962·27-s − 0.176·32-s − 1.04·33-s − 1/3·36-s + 0.657·37-s − 1.28·39-s − 0.904·44-s − 1.76·46-s + 3.50·47-s + 0.144·48-s − 1.42·49-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8809284725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8809284725\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 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
−9.426437543062463645288701294476, −8.525394861615405396000845166993, −8.316812877075225116243141130520, −7.74259453263933502686277981111, −7.29853134793764088505009214313, −7.12027035410652945790309606799, −6.36901426807204320186985434377, −5.48177790941829296116695062663, −5.20592148788576143134616479765, −4.85247836211906827996531873036, −3.85823879659342768603780553916, −2.89341560511568671579379393600, −2.65155040417235216240951299295, −2.23294311234448454772728765221, −0.62279377578968419349957990411,
0.62279377578968419349957990411, 2.23294311234448454772728765221, 2.65155040417235216240951299295, 2.89341560511568671579379393600, 3.85823879659342768603780553916, 4.85247836211906827996531873036, 5.20592148788576143134616479765, 5.48177790941829296116695062663, 6.36901426807204320186985434377, 7.12027035410652945790309606799, 7.29853134793764088505009214313, 7.74259453263933502686277981111, 8.316812877075225116243141130520, 8.525394861615405396000845166993, 9.426437543062463645288701294476
