| L(s) = 1 | + 3-s + 9-s + 6·11-s + 7·13-s + 4·17-s − 5·19-s + 6·23-s + 27-s − 4·29-s − 8·31-s + 6·33-s − 37-s + 7·39-s + 2·41-s + 4·43-s + 8·47-s + 4·51-s − 8·53-s − 5·57-s + 5·61-s − 5·67-s + 6·69-s − 4·71-s − 73-s − 11·79-s + 81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 1.94·13-s + 0.970·17-s − 1.14·19-s + 1.25·23-s + 0.192·27-s − 0.742·29-s − 1.43·31-s + 1.04·33-s − 0.164·37-s + 1.12·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 0.560·51-s − 1.09·53-s − 0.662·57-s + 0.640·61-s − 0.610·67-s + 0.722·69-s − 0.474·71-s − 0.117·73-s − 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.661047731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.661047731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.27071976085418, −14.07743836777769, −13.26689238410232, −12.78890696910269, −12.58246090481955, −11.59174076154959, −11.36847990154072, −10.79570802671541, −10.31271770573750, −9.494369170365131, −9.024824622590339, −8.802718278851416, −8.302584124400001, −7.454883634772667, −7.076841769139766, −6.398285001221768, −5.938517103721956, −5.438489890728697, −4.298014372562774, −4.082904766510771, −3.449384693729496, −3.004259625822082, −1.844854357564287, −1.458688189226408, −0.7684594988340912,
0.7684594988340912, 1.458688189226408, 1.844854357564287, 3.004259625822082, 3.449384693729496, 4.082904766510771, 4.298014372562774, 5.438489890728697, 5.938517103721956, 6.398285001221768, 7.076841769139766, 7.454883634772667, 8.302584124400001, 8.802718278851416, 9.024824622590339, 9.494369170365131, 10.31271770573750, 10.79570802671541, 11.36847990154072, 11.59174076154959, 12.58246090481955, 12.78890696910269, 13.26689238410232, 14.07743836777769, 14.27071976085418
