| L(s) = 1 | + 8.20·2-s + 9·3-s + 35.3·4-s + 29.2·5-s + 73.8·6-s + 27.7·8-s + 81·9-s + 239.·10-s + 377.·11-s + 318.·12-s + 509.·13-s + 263.·15-s − 904.·16-s + 1.60e3·17-s + 664.·18-s + 2.15e3·19-s + 1.03e3·20-s + 3.09e3·22-s − 4.43e3·23-s + 249.·24-s − 2.27e3·25-s + 4.18e3·26-s + 729·27-s + 4.77e3·29-s + 2.15e3·30-s − 7.15e3·31-s − 8.31e3·32-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.10·4-s + 0.522·5-s + 0.837·6-s + 0.153·8-s + 0.333·9-s + 0.758·10-s + 0.940·11-s + 0.638·12-s + 0.836·13-s + 0.301·15-s − 0.883·16-s + 1.34·17-s + 0.483·18-s + 1.37·19-s + 0.578·20-s + 1.36·22-s − 1.74·23-s + 0.0885·24-s − 0.726·25-s + 1.21·26-s + 0.192·27-s + 1.05·29-s + 0.438·30-s − 1.33·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.811641281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.811641281\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 8.20T + 32T^{2} \) |
| 5 | \( 1 - 29.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 377.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 509.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.60e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−12.26859199471980450808551853763, −11.65327168535889613659256876016, −10.08477884285353781060683130388, −9.126428242895399914887692043769, −7.73465321279356183085207015519, −6.32072584444273361134495702546, −5.50562262317752765055544075487, −4.03495956513911780864849964783, −3.18996885715293506350765161367, −1.59018666628297418517816211447,
1.59018666628297418517816211447, 3.18996885715293506350765161367, 4.03495956513911780864849964783, 5.50562262317752765055544075487, 6.32072584444273361134495702546, 7.73465321279356183085207015519, 9.126428242895399914887692043769, 10.08477884285353781060683130388, 11.65327168535889613659256876016, 12.26859199471980450808551853763
