| L(s) = 1 | + 8·2-s − 47.1·3-s + 64·4-s + 395.·5-s − 377.·6-s + 1.40e3·7-s + 512·8-s + 37.6·9-s + 3.16e3·10-s − 3.67e3·11-s − 3.01e3·12-s − 1.22e4·13-s + 1.12e4·14-s − 1.86e4·15-s + 4.09e3·16-s − 2.35e4·17-s + 300.·18-s + 3.95e4·19-s + 2.52e4·20-s − 6.62e4·21-s − 2.94e4·22-s + 2.64e3·23-s − 2.41e4·24-s + 7.79e4·25-s − 9.83e4·26-s + 1.01e5·27-s + 8.99e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s + 1.41·5-s − 0.713·6-s + 1.54·7-s + 0.353·8-s + 0.0171·9-s + 0.999·10-s − 0.832·11-s − 0.504·12-s − 1.55·13-s + 1.09·14-s − 1.42·15-s + 0.250·16-s − 1.16·17-s + 0.0121·18-s + 1.32·19-s + 0.706·20-s − 1.56·21-s − 0.588·22-s + 0.0453·23-s − 0.356·24-s + 0.997·25-s − 1.09·26-s + 0.991·27-s + 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.674471125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.674471125\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 + 47.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 395.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.40e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.35e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.95e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.64e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.18e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.04e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.75e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.09e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.50e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.45e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.79e7T + 8.07e13T^{2} \) |
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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
−10.10400753599642899101702765634, −8.830533754242792770816567408938, −7.65229877099166084907076503783, −6.72762857160128056577883988915, −5.66197897792254238854864223435, −5.10698494641493229434436499239, −4.69190444998912304757557745145, −2.64627666828929645758140809602, −2.01169691526399696709088351941, −0.78192437749901104675321600759,
0.78192437749901104675321600759, 2.01169691526399696709088351941, 2.64627666828929645758140809602, 4.69190444998912304757557745145, 5.10698494641493229434436499239, 5.66197897792254238854864223435, 6.72762857160128056577883988915, 7.65229877099166084907076503783, 8.830533754242792770816567408938, 10.10400753599642899101702765634
