L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 27.6i·5-s + 22.6i·8-s + 78.1·10-s − 68.7i·11-s + 234.·13-s + 64.0·16-s + 232. i·17-s + 358.·19-s − 220. i·20-s − 194.·22-s + 673. i·23-s − 137.·25-s − 661. i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.10i·5-s + 0.353i·8-s + 0.781·10-s − 0.568i·11-s + 1.38·13-s + 0.250·16-s + 0.803i·17-s + 0.992·19-s − 0.552i·20-s − 0.401·22-s + 1.27i·23-s − 0.220·25-s − 0.979i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.018403330\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.018403330\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 27.6iT - 625T^{2} \) |
| 11 | \( 1 + 68.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 234.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 232. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 358.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 673. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 791. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 705.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.37e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 56.5iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 263.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.32e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.05e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.83e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 8.12e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.19e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.05e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.94e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 852.T + 8.85e7T^{2} \) |
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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
−9.875656794462703488622845960897, −8.888307445929097184915437566905, −8.088294983571455070343443461012, −7.08787248108681208122285890833, −6.11861198272688916698875721037, −5.33769057387374299517168186619, −3.65809216660257275255532181840, −3.47630815118703994018280256378, −2.13643993864433542323257677301, −0.999955826430012354608639631053,
0.55097516311904445340400251858, 1.52191534968799766286859003279, 3.22702851838228700754300603338, 4.40671602402071860937377564097, 5.08853699830716864907299639661, 5.95727187847531189601786004732, 6.96701637871699642012504027559, 7.78293823936816693744788089100, 8.873930018503220048687847457691, 8.993842582163377045253659248031