| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 4.53·5-s − 2.50·7-s + 2.82i·8-s + 6.40i·10-s + 10.4i·11-s + 10.6·13-s + 3.54i·14-s + 4.00·16-s + 23.2·17-s − 10.6i·19-s + 9.06·20-s + 14.7·22-s + 2.83·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.906·5-s − 0.358·7-s + 0.353i·8-s + 0.640i·10-s + 0.950i·11-s + 0.822·13-s + 0.253i·14-s + 0.250·16-s + 1.36·17-s − 0.558i·19-s + 0.453·20-s + 0.671·22-s + 0.123·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3013638015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3013638015\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 113 | \( 1 + (27.4 - 109. i)T \) |
| good | 5 | \( 1 + 4.53T + 25T^{2} \) |
| 7 | \( 1 + 2.50T + 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 10.6T + 169T^{2} \) |
| 17 | \( 1 - 23.2T + 289T^{2} \) |
| 19 | \( 1 + 10.6iT - 361T^{2} \) |
| 23 | \( 1 - 2.83T + 529T^{2} \) |
| 29 | \( 1 + 8.27T + 841T^{2} \) |
| 31 | \( 1 + 36.1T + 961T^{2} \) |
| 37 | \( 1 + 11.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.48iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.68iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 44.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 57.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 23.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 92.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 74.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 86.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 120.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 5.77T + 9.40e3T^{2} \) |
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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
−9.270550342580099156757130358712, −8.539790693824852428884504356702, −7.60787358195969576073149339295, −7.12048515032037416751873883064, −5.88155750357423285310804449220, −5.06071695596685484228479177741, −3.93815943168417571199299399164, −3.57234005021498831048945858038, −2.37546260368485861972930074041, −1.16276522106250606379420269716,
0.090089822864139547912226573395, 1.31221337499343534654293809782, 3.26793036057453357123127858412, 3.61255307941969636066635266504, 4.69928291506060034211128118173, 5.81003518071003049504327587942, 6.17348098702087290703313638058, 7.35348938572729197746966384338, 7.87044579651239049665868751563, 8.515619980929242220431502231883
