L(s) = 1 | − 1.41i·2-s + (−1.18 − 2.75i)3-s − 2.00·4-s − 2.43i·5-s + (−3.90 + 1.66i)6-s − 3.65·7-s + 2.82i·8-s + (−6.21 + 6.51i)9-s − 3.43·10-s − 14.0i·11-s + (2.36 + 5.51i)12-s + 3.60·13-s + 5.17i·14-s + (−6.70 + 2.87i)15-s + 4.00·16-s − 7.30i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.393 − 0.919i)3-s − 0.500·4-s − 0.486i·5-s + (−0.650 + 0.278i)6-s − 0.522·7-s + 0.353i·8-s + (−0.690 + 0.723i)9-s − 0.343·10-s − 1.27i·11-s + (0.196 + 0.459i)12-s + 0.277·13-s + 0.369i·14-s + (−0.447 + 0.191i)15-s + 0.250·16-s − 0.429i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.182638 - 0.890531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.182638 - 0.890531i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (1.18 + 2.75i)T \) |
| 13 | \( 1 - 3.60T \) |
good | 5 | \( 1 + 2.43iT - 25T^{2} \) |
| 7 | \( 1 + 3.65T + 49T^{2} \) |
| 11 | \( 1 + 14.0iT - 121T^{2} \) |
| 17 | \( 1 + 7.30iT - 289T^{2} \) |
| 19 | \( 1 - 24.9T + 361T^{2} \) |
| 23 | \( 1 + 33.9iT - 529T^{2} \) |
| 29 | \( 1 - 23.1iT - 841T^{2} \) |
| 31 | \( 1 + 15.4T + 961T^{2} \) |
| 37 | \( 1 - 36.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 29.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 33.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 10.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 104.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 69.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 63.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 134.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 67.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 142.T + 9.40e3T^{2} \) |
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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
−13.41035640204663186318382609455, −12.64321971124436042729442426061, −11.64272451795678978874547608949, −10.71792979253032163722570653130, −9.167416700847780340214868028997, −8.099249031893531538068732378992, −6.49710898453677236945030613803, −5.17290639180228298831341555366, −3.03835876027900824368756281384, −0.859420984618529308017270516932,
3.55663951286357636258122413479, 5.05199813136559568072633791581, 6.33382376468186618832064204050, 7.58009291602916366450300607335, 9.322623933953601505718376355349, 9.965158845887619543884419445394, 11.26796247098523684467634049228, 12.53204221774756140542501502367, 13.89038821404801850469196625998, 15.02008850380615797736824627596