| L(s) = 1 | + 2-s + 4-s + 4.38·7-s + 8-s − 2·11-s − 13-s + 4.38·14-s + 16-s + 5.86·17-s − 0.973·19-s − 2·22-s + 7.79·23-s − 26-s + 4.38·28-s − 0.973·29-s − 1.79·31-s + 32-s + 5.86·34-s + 0.591·37-s − 0.973·38-s − 4.81·41-s − 4.68·43-s − 2·44-s + 7.79·46-s + 0.381·47-s + 12.1·49-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.65·7-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 1.17·14-s + 0.250·16-s + 1.42·17-s − 0.223·19-s − 0.426·22-s + 1.62·23-s − 0.196·26-s + 0.828·28-s − 0.180·29-s − 0.321·31-s + 0.176·32-s + 1.00·34-s + 0.0972·37-s − 0.157·38-s − 0.752·41-s − 0.714·43-s − 0.301·44-s + 1.14·46-s + 0.0556·47-s + 1.74·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.175495400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.175495400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 + 0.973T + 19T^{2} \) |
| 23 | \( 1 - 7.79T + 23T^{2} \) |
| 29 | \( 1 + 0.973T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 - 0.591T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 0.381T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 - 0.973T + 59T^{2} \) |
| 61 | \( 1 + 0.817T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 + 0.973T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.981267485804404022571294126993, −7.42566328450048093725462329928, −6.74541771013221189530218980370, −5.57819601364454800205672567273, −5.21039040137060604947210738225, −4.66900664890338250431758766849, −3.71122162849884799228237559385, −2.85285923543981849146668516826, −1.93158311479120798124281007810, −1.04440546448079732842979602719,
1.04440546448079732842979602719, 1.93158311479120798124281007810, 2.85285923543981849146668516826, 3.71122162849884799228237559385, 4.66900664890338250431758766849, 5.21039040137060604947210738225, 5.57819601364454800205672567273, 6.74541771013221189530218980370, 7.42566328450048093725462329928, 7.981267485804404022571294126993
