| L(s) = 1 | + (0.735 + 2.01i)2-s + (−1.28 + 1.16i)3-s + (−2.00 + 1.68i)4-s + (0.0677 + 0.384i)5-s + (−3.29 − 1.73i)6-s + (0.618 + 2.57i)7-s + (−1.15 − 0.666i)8-s + (0.295 − 2.98i)9-s + (−0.726 + 0.419i)10-s + (−4.77 − 0.842i)11-s + (0.617 − 4.49i)12-s + (1.96 − 5.38i)13-s + (−4.74 + 3.14i)14-s + (−0.534 − 0.414i)15-s + (−0.412 + 2.34i)16-s + (3.57 + 6.18i)17-s + ⋯ |
| L(s) = 1 | + (0.519 + 1.42i)2-s + (−0.741 + 0.671i)3-s + (−1.00 + 0.841i)4-s + (0.0303 + 0.171i)5-s + (−1.34 − 0.709i)6-s + (0.233 + 0.972i)7-s + (−0.407 − 0.235i)8-s + (0.0983 − 0.995i)9-s + (−0.229 + 0.132i)10-s + (−1.44 − 0.254i)11-s + (0.178 − 1.29i)12-s + (0.543 − 1.49i)13-s + (−1.26 + 0.839i)14-s + (−0.137 − 0.107i)15-s + (−0.103 + 0.585i)16-s + (0.866 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0332997 + 1.16144i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0332997 + 1.16144i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 1.16i)T \) |
| 7 | \( 1 + (-0.618 - 2.57i)T \) |
| good | 2 | \( 1 + (-0.735 - 2.01i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0677 - 0.384i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (4.77 + 0.842i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.96 + 5.38i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.57 - 6.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.398 - 0.230i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.358 - 0.427i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 4.76i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.14 - 2.55i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.07 + 1.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.917 - 0.334i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.16 + 6.58i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.83 + 2.38i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 3.13iT - 53T^{2} \) |
| 59 | \( 1 + (0.715 + 4.05i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.89 + 9.40i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.30 - 2.29i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.08 + 3.51i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.55 + 3.78i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.818 - 0.297i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.78 + 2.46i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.65 + 4.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 1.90i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.98098824152242636438920221072, −12.45489859481304630855850733466, −10.84993576545518065407829786566, −10.31903811766691042989050425101, −8.589691055600064972910493344196, −7.959142666735941349823895112622, −6.44907699612121144561571794160, −5.50279447537787076367729350077, −5.15502484072759500122712076607, −3.38085059951216313799174021029,
1.07229928550581574744042459780, 2.61836206281162825733754999860, 4.37907384393794960606399939458, 5.20193291700446762215405757122, 6.89346899164626614103809673502, 7.83396206625473038126817473579, 9.628757018830650044472385145907, 10.52528053881517622977611222653, 11.32942911501424510975175137447, 11.92038281947445320966041659602
