| L(s) = 1 | + 3-s + 2·5-s + 9-s + 8·11-s + 2·15-s − 4·17-s − 25-s + 27-s + 8·33-s − 8·43-s + 2·45-s − 14·49-s − 4·51-s + 4·53-s + 16·55-s + 8·59-s − 4·61-s + 8·67-s + 16·71-s − 75-s + 81-s − 8·85-s + 8·99-s − 32·103-s − 4·109-s − 36·113-s + 26·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 2.41·11-s + 0.516·15-s − 0.970·17-s − 1/5·25-s + 0.192·27-s + 1.39·33-s − 1.21·43-s + 0.298·45-s − 2·49-s − 0.560·51-s + 0.549·53-s + 2.15·55-s + 1.04·59-s − 0.512·61-s + 0.977·67-s + 1.89·71-s − 0.115·75-s + 1/9·81-s − 0.867·85-s + 0.804·99-s − 3.15·103-s − 0.383·109-s − 3.38·113-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401538722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401538722\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.703395692631468853448056916432, −9.298977106556460794427243537385, −8.738054065213463721777507480749, −8.362573092386573492350464456460, −7.77541900804615864403738320707, −6.88451958096732153826690211844, −6.50812189570177474416634496690, −6.46625691967373786218224646311, −5.48498394296229820428473072147, −4.93489655467566708723346193761, −3.96665802174365669665598466984, −3.89741909491206350829507060388, −2.86970638476716148313728818699, −1.97363471421369880359461961552, −1.38738099288330045270314903054,
1.38738099288330045270314903054, 1.97363471421369880359461961552, 2.86970638476716148313728818699, 3.89741909491206350829507060388, 3.96665802174365669665598466984, 4.93489655467566708723346193761, 5.48498394296229820428473072147, 6.46625691967373786218224646311, 6.50812189570177474416634496690, 6.88451958096732153826690211844, 7.77541900804615864403738320707, 8.362573092386573492350464456460, 8.738054065213463721777507480749, 9.298977106556460794427243537385, 9.703395692631468853448056916432
