| L(s) = 1 | + 1.41·2-s + 2.00·4-s + (3.53 + 3.53i)5-s − 4i·7-s + 2.82·8-s + (5.00 + 5.00i)10-s + 11.3i·11-s − 18i·13-s − 5.65i·14-s + 4.00·16-s + 1.41·17-s − 24·19-s + (7.07 + 7.07i)20-s + 16.0i·22-s − 39.5·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + (0.707 + 0.707i)5-s − 0.571i·7-s + 0.353·8-s + (0.500 + 0.500i)10-s + 1.02i·11-s − 1.38i·13-s − 0.404i·14-s + 0.250·16-s + 0.0831·17-s − 1.26·19-s + (0.353 + 0.353i)20-s + 0.727i·22-s − 1.72·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.01884 + 0.171933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01884 + 0.171933i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.53 - 3.53i)T \) |
| good | 7 | \( 1 + 4iT - 49T^{2} \) |
| 11 | \( 1 - 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 18iT - 169T^{2} \) |
| 17 | \( 1 - 1.41T + 289T^{2} \) |
| 19 | \( 1 + 24T + 361T^{2} \) |
| 23 | \( 1 + 39.5T + 529T^{2} \) |
| 29 | \( 1 + 38.1iT - 841T^{2} \) |
| 31 | \( 1 - 4T + 961T^{2} \) |
| 37 | \( 1 - 56iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 24.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 80iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 28.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.24T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 110T + 3.72e3T^{2} \) |
| 67 | \( 1 + 32iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 36T + 6.24e3T^{2} \) |
| 83 | \( 1 + 5.65T + 6.88e3T^{2} \) |
| 89 | \( 1 + 57.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14iT - 9.40e3T^{2} \) |
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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
−13.81979767807129398173510711589, −13.04494609541273097859505013509, −11.93174326405436571178010205166, −10.45620760212290489180462808261, −10.03684431987681018907543437181, −7.998539471986806291708482987002, −6.78813006330770941475977453949, −5.66658596079082237225997388622, −4.06702618442472212093837562745, −2.34793613464742305399738959012,
2.09090799867025304411597792322, 4.14047370860834566960260784580, 5.57223505291740036020578767443, 6.48592289266489982262080391305, 8.374280604907503836967485209177, 9.333808433137071094798133812413, 10.78360060118552097773527580517, 11.96350026068277339535107441860, 12.81043719432199669432693784787, 13.89798560656127686530826212184
