L(s) = 1 | + 1.30i·2-s + (−1.82 + 2.38i)3-s + 2.29·4-s − 7.37i·5-s + (−3.11 − 2.38i)6-s − 2.64·7-s + 8.22i·8-s + (−2.35 − 8.68i)9-s + 9.64·10-s + 2.61i·11-s + (−4.17 + 5.45i)12-s − 6.35·13-s − 3.45i·14-s + (17.5 + 13.4i)15-s − 1.58·16-s + 12.1i·17-s + ⋯ |
L(s) = 1 | + 0.653i·2-s + (−0.607 + 0.794i)3-s + 0.572·4-s − 1.47i·5-s + (−0.519 − 0.397i)6-s − 0.377·7-s + 1.02i·8-s + (−0.261 − 0.965i)9-s + 0.964·10-s + 0.237i·11-s + (−0.348 + 0.454i)12-s − 0.488·13-s − 0.247i·14-s + (1.17 + 0.896i)15-s − 0.0989·16-s + 0.714i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.742454 + 0.366798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.742454 + 0.366798i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.82 - 2.38i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 - 1.30iT - 4T^{2} \) |
| 5 | \( 1 + 7.37iT - 25T^{2} \) |
| 11 | \( 1 - 2.61iT - 121T^{2} \) |
| 13 | \( 1 + 6.35T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 + 10.2T + 361T^{2} \) |
| 23 | \( 1 + 4.30iT - 529T^{2} \) |
| 29 | \( 1 + 17.3iT - 841T^{2} \) |
| 31 | \( 1 - 39.2T + 961T^{2} \) |
| 37 | \( 1 - 41.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 89.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.1T + 9.40e3T^{2} \) |
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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
−17.39330515913206427519390381782, −16.75110468915674496519824156456, −15.90918020796956462301426052915, −14.87537043484530243946296448027, −12.77676763769673097375325456987, −11.61055449222500752374842862982, −9.912622864366688966299283425911, −8.365753564601864572636346836896, −6.23009728629170269715395110239, −4.76189951599963393312237247570,
2.72394693730293101925304774111, 6.35514300323772687842844043143, 7.31850106587427905497581587199, 10.22044971433123207637602782910, 11.18926705230815417960428913541, 12.21212455621933931337508124739, 13.70199429714925516254534406272, 15.23636172881219556133467040338, 16.69668689372146167554062418988, 18.21700636530664691977716595000