| L(s) = 1 | − 3-s − 3·4-s − 4·5-s + 4·7-s + 9-s + 3·12-s + 4·15-s + 5·16-s − 4·17-s + 12·20-s − 4·21-s + 2·25-s − 27-s − 12·28-s − 16·35-s − 3·36-s + 12·37-s − 4·41-s − 4·45-s + 16·47-s − 5·48-s + 9·49-s + 4·51-s − 8·59-s − 12·60-s + 4·63-s − 3·64-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 1.78·5-s + 1.51·7-s + 1/3·9-s + 0.866·12-s + 1.03·15-s + 5/4·16-s − 0.970·17-s + 2.68·20-s − 0.872·21-s + 2/5·25-s − 0.192·27-s − 2.26·28-s − 2.70·35-s − 1/2·36-s + 1.97·37-s − 0.624·41-s − 0.596·45-s + 2.33·47-s − 0.721·48-s + 9/7·49-s + 0.560·51-s − 1.04·59-s − 1.54·60-s + 0.503·63-s − 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.975261543279013968212899152873, −8.402747192524386456742003548704, −8.092649848612792766954418771551, −7.55113610427119386648970831546, −7.46995816184461358823952634116, −6.55118736577760269645886929040, −5.83278296173294764308357143188, −5.31622482563829752699303340190, −4.65970646396455232760718697930, −4.28499925734034658438358048834, −4.20407903283329927801590362766, −3.43632280535116392331032608080, −2.26752106768048411048763527476, −1.03326922256959689494698211571, 0,
1.03326922256959689494698211571, 2.26752106768048411048763527476, 3.43632280535116392331032608080, 4.20407903283329927801590362766, 4.28499925734034658438358048834, 4.65970646396455232760718697930, 5.31622482563829752699303340190, 5.83278296173294764308357143188, 6.55118736577760269645886929040, 7.46995816184461358823952634116, 7.55113610427119386648970831546, 8.092649848612792766954418771551, 8.402747192524386456742003548704, 8.975261543279013968212899152873
