| L(s) = 1 | − 2.10·2-s + 3-s + 2.43·4-s + 5-s − 2.10·6-s + 3.31·7-s − 0.923·8-s + 9-s − 2.10·10-s − 1.00·11-s + 2.43·12-s − 13-s − 6.99·14-s + 15-s − 2.93·16-s + 2.78·17-s − 2.10·18-s + 7.42·19-s + 2.43·20-s + 3.31·21-s + 2.10·22-s + 4.76·23-s − 0.923·24-s + 25-s + 2.10·26-s + 27-s + 8.09·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.21·4-s + 0.447·5-s − 0.860·6-s + 1.25·7-s − 0.326·8-s + 0.333·9-s − 0.666·10-s − 0.301·11-s + 0.703·12-s − 0.277·13-s − 1.86·14-s + 0.258·15-s − 0.732·16-s + 0.675·17-s − 0.496·18-s + 1.70·19-s + 0.545·20-s + 0.724·21-s + 0.449·22-s + 0.992·23-s − 0.188·24-s + 0.200·25-s + 0.413·26-s + 0.192·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764847106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764847106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 7.54T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 + 2.81T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5.25T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.02T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 2.35T + 97T^{2} \) |
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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
−8.137955708102718068293623570866, −7.53167062438837167256503125665, −7.22007858727208198932446528563, −6.14359361807784623479267714367, −5.04588000100920572973499043496, −4.70353516838906079359654542906, −3.25298006715522050428427042056, −2.50744865011794351245526448605, −1.50591302538739162198963743749, −0.969251747791951477470053944063,
0.969251747791951477470053944063, 1.50591302538739162198963743749, 2.50744865011794351245526448605, 3.25298006715522050428427042056, 4.70353516838906079359654542906, 5.04588000100920572973499043496, 6.14359361807784623479267714367, 7.22007858727208198932446528563, 7.53167062438837167256503125665, 8.137955708102718068293623570866
