| L(s) = 1 | + 3.56i·3-s − 2.91·5-s − 2.95·7-s − 3.72·9-s − 21.8·11-s − 10.0i·13-s − 10.3i·15-s + 8.21·17-s − 10.5i·21-s + 12.5·23-s − 16.5·25-s + 18.8i·27-s + 21.5i·29-s − 41.3i·31-s − 78.0i·33-s + ⋯ |
| L(s) = 1 | + 1.18i·3-s − 0.582·5-s − 0.421·7-s − 0.413·9-s − 1.98·11-s − 0.775i·13-s − 0.692i·15-s + 0.483·17-s − 0.501i·21-s + 0.544·23-s − 0.660·25-s + 0.696i·27-s + 0.741i·29-s − 1.33i·31-s − 2.36i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.035648139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035648139\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.56iT - 9T^{2} \) |
| 5 | \( 1 + 2.91T + 25T^{2} \) |
| 7 | \( 1 + 2.95T + 49T^{2} \) |
| 11 | \( 1 + 21.8T + 121T^{2} \) |
| 13 | \( 1 + 10.0iT - 169T^{2} \) |
| 17 | \( 1 - 8.21T + 289T^{2} \) |
| 23 | \( 1 - 12.5T + 529T^{2} \) |
| 29 | \( 1 - 21.5iT - 841T^{2} \) |
| 31 | \( 1 + 41.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 74.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 31.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 54.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 61.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 26.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 144. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 2.17T + 6.88e3T^{2} \) |
| 89 | \( 1 + 19.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 110. iT - 9.40e3T^{2} \) |
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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
−9.631836513294773479754951434456, −8.532130437898374367383333532715, −7.83786752890743947654728630205, −7.13150483772587737334481923357, −5.63914394523175274969039483079, −5.24211328834402116925709457378, −4.20546881288018518141645286343, −3.38661646121585314703139894468, −2.53700072900378558145373536839, −0.42546715560738696731489445759,
0.74318925319370702328852576260, 2.12327664408002404959984441105, 2.96722001018464922249151109296, 4.20388495663339585186515592218, 5.29005362635045314566009964273, 6.16575465051691890017777372041, 7.10702767037491932149594388992, 7.65401032592005282417615076200, 8.180327264962982976728411587020, 9.229267133798620529762509798145
