| L(s) = 1 | + 1.41i·2-s + (−2.73 + 1.23i)3-s − 2.00·4-s − 2.23i·5-s + (−1.75 − 3.86i)6-s + 4.14·7-s − 2.82i·8-s + (5.93 − 6.76i)9-s + 3.16·10-s + 18.2i·11-s + (5.46 − 2.47i)12-s − 8.45·13-s + 5.85i·14-s + (2.76 + 6.11i)15-s + 4.00·16-s − 14.2i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.910 + 0.412i)3-s − 0.500·4-s − 0.447i·5-s + (−0.291 − 0.644i)6-s + 0.591·7-s − 0.353i·8-s + (0.659 − 0.751i)9-s + 0.316·10-s + 1.65i·11-s + (0.455 − 0.206i)12-s − 0.650·13-s + 0.418i·14-s + (0.184 + 0.407i)15-s + 0.250·16-s − 0.839i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4552512435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4552512435\) |
| \(L(2)\) |
|
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 | \( 1 + (2.73 - 1.23i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 - 4.79iT \) |
| good | 7 | \( 1 - 4.14T + 49T^{2} \) |
| 11 | \( 1 - 18.2iT - 121T^{2} \) |
| 13 | \( 1 + 8.45T + 169T^{2} \) |
| 17 | \( 1 + 14.2iT - 289T^{2} \) |
| 19 | \( 1 + 25.5T + 361T^{2} \) |
| 29 | \( 1 + 30.0iT - 841T^{2} \) |
| 31 | \( 1 + 20.5T + 961T^{2} \) |
| 37 | \( 1 - 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 13.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 79.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 75.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 132. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 23.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 116. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 82.3T + 9.40e3T^{2} \) |
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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
−9.830276899516962088820114440290, −9.504064317491172323214103076894, −8.252936193448900055404871172159, −7.31120582204567048183165825984, −6.60559529434233778470708175956, −5.42916310720810103230378059668, −4.73783521810312263189372970740, −4.14411488108061252502162689472, −1.97466037742006553006989509886, −0.19746238260349294005688942914,
1.24861165452088250269404064891, 2.52020457899974731750123163265, 3.87360537825986549738027692081, 4.97305497455280810595477126596, 5.92596087950457819404765928474, 6.71333070064094598229573419540, 7.983545228605811407838238122743, 8.575544197396910621562660550690, 9.927520517905980362601168903266, 10.75138583751644753291161801313
