| L(s) = 1 | + 1.13i·2-s + (−2.53 + 1.60i)3-s + 2.70·4-s + 4.29i·5-s + (−1.82 − 2.89i)6-s − 9.25·7-s + 7.63i·8-s + (3.86 − 8.12i)9-s − 4.89·10-s + 15.8i·11-s + (−6.84 + 4.32i)12-s + 10.0·13-s − 10.5i·14-s + (−6.88 − 10.9i)15-s + 2.09·16-s − 27.1i·17-s + ⋯ |
| L(s) = 1 | + 0.569i·2-s + (−0.845 + 0.534i)3-s + 0.675·4-s + 0.859i·5-s + (−0.304 − 0.481i)6-s − 1.32·7-s + 0.954i·8-s + (0.429 − 0.903i)9-s − 0.489·10-s + 1.44i·11-s + (−0.570 + 0.360i)12-s + 0.771·13-s − 0.753i·14-s + (−0.459 − 0.726i)15-s + 0.131·16-s − 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.482700 + 0.876045i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482700 + 0.876045i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.53 - 1.60i)T \) |
| 23 | \( 1 + 4.79iT \) |
| good | 2 | \( 1 - 1.13iT - 4T^{2} \) |
| 5 | \( 1 - 4.29iT - 25T^{2} \) |
| 7 | \( 1 + 9.25T + 49T^{2} \) |
| 11 | \( 1 - 15.8iT - 121T^{2} \) |
| 13 | \( 1 - 10.0T + 169T^{2} \) |
| 17 | \( 1 + 27.1iT - 289T^{2} \) |
| 19 | \( 1 - 27.5T + 361T^{2} \) |
| 29 | \( 1 - 19.8iT - 841T^{2} \) |
| 31 | \( 1 - 12.5T + 961T^{2} \) |
| 37 | \( 1 + 20.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 7.46iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 25.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 29.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 38.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 14.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 139.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 5.81iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 48.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 66.6T + 9.40e3T^{2} \) |
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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
−15.33407043393460825331492035528, −14.04513710887356026457534741486, −12.42254908434757531758122030345, −11.49288904178552527886214460941, −10.36172163557904049406995669286, −9.476566279725977675513817785747, −7.09710618504980179682753755618, −6.71151665768343898981039352345, −5.28365364188975002587427451893, −3.15931393937891969078153888837,
1.06200692506277104737476807478, 3.43882822669262336197125282354, 5.77287180958336272622669007151, 6.57059784280059547856626358769, 8.262087969273696928524483743345, 9.875513528777749601639272061312, 10.98920103382193270275952537173, 11.91490327988124169116480546213, 12.89574506716231906443163726080, 13.46812043853554779833058242503
