| L(s) = 1 | + 3.43i·3-s + (−3.90 − 3.11i)5-s − 2.82·7-s − 2.81·9-s + (7.81 + 7.74i)11-s + 3.98·13-s + (10.7 − 13.4i)15-s + 23.2·17-s − 33.1i·19-s − 9.72i·21-s + 2.16i·23-s + (5.53 + 24.3i)25-s + 21.2i·27-s + 11.5i·29-s − 36.7·31-s + ⋯ |
| L(s) = 1 | + 1.14i·3-s + (−0.781 − 0.623i)5-s − 0.404·7-s − 0.312·9-s + (0.710 + 0.703i)11-s + 0.306·13-s + (0.714 − 0.895i)15-s + 1.36·17-s − 1.74i·19-s − 0.462i·21-s + 0.0941i·23-s + (0.221 + 0.975i)25-s + 0.787i·27-s + 0.397i·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.499049245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499049245\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.90 + 3.11i)T \) |
| 11 | \( 1 + (-7.81 - 7.74i)T \) |
| good | 3 | \( 1 - 3.43iT - 9T^{2} \) |
| 7 | \( 1 + 2.82T + 49T^{2} \) |
| 13 | \( 1 - 3.98T + 169T^{2} \) |
| 17 | \( 1 - 23.2T + 289T^{2} \) |
| 19 | \( 1 + 33.1iT - 361T^{2} \) |
| 23 | \( 1 - 2.16iT - 529T^{2} \) |
| 29 | \( 1 - 11.5iT - 841T^{2} \) |
| 31 | \( 1 + 36.7T + 961T^{2} \) |
| 37 | \( 1 - 17.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 45.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 63.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 44.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 38.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 54.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 56.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 101. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 15.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 76.2iT - 9.40e3T^{2} \) |
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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
−9.973538308235342342740775395820, −9.366780140022409723000777653187, −8.795097775582681148165270708503, −7.63221033117598187977322250090, −6.85860783600159143080064860348, −5.50576546967480622069162088460, −4.66281305375000253261003887052, −3.97459198997672009813444270896, −3.08367802996651315288453341477, −1.12437914452120170645594115221,
0.60198046082485043388583128475, 1.84707423226630560173386717166, 3.33748744852453444789723044858, 3.91442027954074337249336098082, 5.73222913193860688320027649635, 6.31492393603500901609339430214, 7.29703652769289583953344444651, 7.78910323175899833121373591087, 8.614837303688192325582309016882, 9.784572742292195051386106449970
