L(s) = 1 | + (1.73 + 0.986i)2-s − 1.31i·3-s + (2.05 + 3.43i)4-s + 2.48·5-s + (1.30 − 2.29i)6-s − 0.383i·7-s + (0.180 + 7.99i)8-s + 7.25·9-s + (4.32 + 2.45i)10-s − 1.10i·11-s + (4.53 − 2.70i)12-s − 3.59·13-s + (0.378 − 0.666i)14-s − 3.28i·15-s + (−7.57 + 14.0i)16-s − 9.84·17-s + ⋯ |
L(s) = 1 | + (0.869 + 0.493i)2-s − 0.439i·3-s + (0.512 + 0.858i)4-s + 0.497·5-s + (0.217 − 0.382i)6-s − 0.0547i·7-s + (0.0226 + 0.999i)8-s + 0.806·9-s + (0.432 + 0.245i)10-s − 0.100i·11-s + (0.377 − 0.225i)12-s − 0.276·13-s + (0.0270 − 0.0476i)14-s − 0.218i·15-s + (−0.473 + 0.880i)16-s − 0.579·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.30071 + 0.635099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.30071 + 0.635099i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.73 - 0.986i)T \) |
| 31 | \( 1 - 5.56iT \) |
good | 3 | \( 1 + 1.31iT - 9T^{2} \) |
| 5 | \( 1 - 2.48T + 25T^{2} \) |
| 7 | \( 1 + 0.383iT - 49T^{2} \) |
| 11 | \( 1 + 1.10iT - 121T^{2} \) |
| 13 | \( 1 + 3.59T + 169T^{2} \) |
| 17 | \( 1 + 9.84T + 289T^{2} \) |
| 19 | \( 1 + 9.77iT - 361T^{2} \) |
| 23 | \( 1 + 10.9iT - 529T^{2} \) |
| 29 | \( 1 + 25.5T + 841T^{2} \) |
| 37 | \( 1 + 32.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 2.53T + 1.68e3T^{2} \) |
| 43 | \( 1 + 38.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.76iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 27.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 126. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 45.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 114. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 78.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 3.38T + 7.92e3T^{2} \) |
| 97 | \( 1 - 42.4T + 9.40e3T^{2} \) |
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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
−13.31420274829137271460017953586, −12.57280015177178580891808892187, −11.51522986178097624005154516360, −10.23469848295199445311010206060, −8.797617169747064407776730982986, −7.44750725066181442933801120372, −6.63848491399463465100774845407, −5.39233270476703400531858370230, −4.04551894314279822540592575028, −2.20239619157451542023743055783,
1.91295263483788087806046281884, 3.70195920216630553089210837145, 4.89031339782318791610604851955, 6.08481088233902913316542408982, 7.38321856437703459644209610064, 9.322404319424320731540017403711, 10.07933809927581316380523034896, 11.03469085605675676856254586897, 12.18777218617624912681465538150, 13.11281907506872428879490439372