L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 2·13-s + 4·17-s + 3·19-s + 21-s − 23-s − 5·25-s − 27-s − 6·29-s + 2·31-s + 33-s − 2·37-s − 2·39-s + 41-s + 8·43-s + 5·47-s + 49-s − 4·51-s + 3·53-s − 3·57-s − 5·59-s + 13·61-s − 63-s + 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.970·17-s + 0.688·19-s + 0.218·21-s − 0.208·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.156·41-s + 1.21·43-s + 0.729·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s − 0.397·57-s − 0.650·59-s + 1.66·61-s − 0.125·63-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.434066730\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434066730\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65615669011942098304229515647, −7.29757831676853653141972161827, −6.33848484629361277428794008313, −5.69940678563817455326413292079, −5.33616679689971623468849699076, −4.21419228565879097102747132494, −3.64025377614519363547254889260, −2.73333521245439342564505080597, −1.64583894647173953166610197524, −0.63021108892516670440197132311,
0.63021108892516670440197132311, 1.64583894647173953166610197524, 2.73333521245439342564505080597, 3.64025377614519363547254889260, 4.21419228565879097102747132494, 5.33616679689971623468849699076, 5.69940678563817455326413292079, 6.33848484629361277428794008313, 7.29757831676853653141972161827, 7.65615669011942098304229515647