| L(s) = 1 | + (−0.645 − 1.89i)2-s + (−3.16 + 2.44i)4-s + 2.38·5-s + 12.3i·7-s + (6.67 + 4.41i)8-s + (−1.53 − 4.50i)10-s + 9.15i·11-s + 0.940·13-s + (23.4 − 7.98i)14-s + (4.05 − 15.4i)16-s − 27.1·17-s − 4.35i·19-s + (−7.54 + 5.82i)20-s + (17.3 − 5.90i)22-s − 13.8i·23-s + ⋯ |
| L(s) = 1 | + (−0.322 − 0.946i)2-s + (−0.791 + 0.611i)4-s + 0.476·5-s + 1.76i·7-s + (0.833 + 0.552i)8-s + (−0.153 − 0.450i)10-s + 0.831i·11-s + 0.0723·13-s + (1.67 − 0.570i)14-s + (0.253 − 0.967i)16-s − 1.59·17-s − 0.229i·19-s + (−0.377 + 0.291i)20-s + (0.787 − 0.268i)22-s − 0.600i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4453992027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4453992027\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.645 + 1.89i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 2.38T + 25T^{2} \) |
| 7 | \( 1 - 12.3iT - 49T^{2} \) |
| 11 | \( 1 - 9.15iT - 121T^{2} \) |
| 13 | \( 1 - 0.940T + 169T^{2} \) |
| 17 | \( 1 + 27.1T + 289T^{2} \) |
| 23 | \( 1 + 13.8iT - 529T^{2} \) |
| 29 | \( 1 + 49.8T + 841T^{2} \) |
| 31 | \( 1 + 24.6iT - 961T^{2} \) |
| 37 | \( 1 + 27.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 19.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 33.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.48iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 88.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 51.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.62iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 31.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 159.T + 9.40e3T^{2} \) |
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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
−10.59361203491361825375235247954, −9.563105907504906942605455223530, −9.089168122380135456189152902466, −8.417204579711282300387414645123, −7.19156326304678072960576859178, −5.92837466260221534180721864453, −5.06206579033477147271337260167, −3.94224502880493510807032175733, −2.33645040958249264977419021739, −2.09250196310594999724413324499,
0.17233094655408973739994677113, 1.53827273205299578867579758316, 3.65728423823541177081664628020, 4.48489727622649815152436207495, 5.66348325461546631702918848511, 6.54416334958034367177295638076, 7.30289114528946010034003938820, 8.065855645230273316487544213661, 9.110015078584479016952295065742, 9.800214434541318682313500357171
