| L(s) = 1 | − 4·3-s − 4·4-s + 6·5-s + 6·9-s + 3·11-s + 16·12-s − 24·15-s + 12·16-s − 24·20-s + 17·25-s + 4·27-s − 8·31-s − 12·33-s − 24·36-s + 4·37-s − 12·44-s + 36·45-s − 6·47-s − 48·48-s − 13·49-s + 24·53-s + 18·55-s − 12·59-s + 96·60-s − 32·64-s − 8·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2·4-s + 2.68·5-s + 2·9-s + 0.904·11-s + 4.61·12-s − 6.19·15-s + 3·16-s − 5.36·20-s + 17/5·25-s + 0.769·27-s − 1.43·31-s − 2.08·33-s − 4·36-s + 0.657·37-s − 1.80·44-s + 5.36·45-s − 0.875·47-s − 6.92·48-s − 1.85·49-s + 3.29·53-s + 2.42·55-s − 1.56·59-s + 12.3·60-s − 4·64-s − 0.977·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5640125534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5640125534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.24320445923116880350231460909, −9.789089988406658813136176124228, −9.180168549070062598241568188033, −9.100215838775092853341574624634, −8.428127489946401133860742721663, −7.42654336671663498602630753206, −6.39084306102520193435942181609, −6.22903281660809054167680244992, −5.78502585044634421931712450323, −5.30410557441328121366051727290, −5.03912355415234199771433602492, −4.42264240278191716580392851586, −3.36608228640882686115534658030, −1.83661141375158432347083933452, −0.807781280276512774607779040405,
0.807781280276512774607779040405, 1.83661141375158432347083933452, 3.36608228640882686115534658030, 4.42264240278191716580392851586, 5.03912355415234199771433602492, 5.30410557441328121366051727290, 5.78502585044634421931712450323, 6.22903281660809054167680244992, 6.39084306102520193435942181609, 7.42654336671663498602630753206, 8.428127489946401133860742721663, 9.100215838775092853341574624634, 9.180168549070062598241568188033, 9.789089988406658813136176124228, 10.24320445923116880350231460909
