| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s + 3·11-s − 13-s − 3·15-s − 17-s + 7·19-s − 21-s + 23-s + 4·25-s − 27-s + 10·29-s + 4·31-s − 3·33-s + 3·35-s + 10·37-s + 39-s + 3·41-s + 11·43-s + 3·45-s − 8·47-s + 49-s + 51-s + 4·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.774·15-s − 0.242·17-s + 1.60·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.522·33-s + 0.507·35-s + 1.64·37-s + 0.160·39-s + 0.468·41-s + 1.67·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.549·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.572216455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.572216455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.57829586672837, −14.84194509756459, −14.26870839949289, −13.78014882327901, −13.60910747339306, −12.59714926891574, −12.34327462109611, −11.57568395180430, −11.21680985736470, −10.52031424513650, −9.839309837622952, −9.573022714102662, −9.049978328185992, −8.223138620075912, −7.564779885681911, −6.875618699798949, −6.202691253748164, −5.978224328854640, −5.086338104985545, −4.734468432131598, −3.926976304599646, −2.880447738003235, −2.340902489858167, −1.274611651805787, −0.9406753868897240,
0.9406753868897240, 1.274611651805787, 2.340902489858167, 2.880447738003235, 3.926976304599646, 4.734468432131598, 5.086338104985545, 5.978224328854640, 6.202691253748164, 6.875618699798949, 7.564779885681911, 8.223138620075912, 9.049978328185992, 9.573022714102662, 9.839309837622952, 10.52031424513650, 11.21680985736470, 11.57568395180430, 12.34327462109611, 12.59714926891574, 13.60910747339306, 13.78014882327901, 14.26870839949289, 14.84194509756459, 15.57829586672837
