| L(s) = 1 | + (2.43 − 0.612i)2-s + (−0.667 − 0.883i)3-s + (3.80 − 2.04i)4-s + (−2.73 + 3.19i)5-s + (−2.16 − 1.74i)6-s + (−1.82 + 0.582i)7-s + (4.30 − 3.92i)8-s + (0.485 − 1.70i)9-s + (−4.70 + 9.45i)10-s + (0.844 + 0.870i)11-s + (−4.34 − 1.99i)12-s + (−2.75 − 2.50i)13-s + (−4.10 + 2.54i)14-s + (4.64 + 0.286i)15-s + (3.31 − 4.99i)16-s + (0.0151 − 0.0121i)17-s + ⋯ |
| L(s) = 1 | + (1.72 − 0.433i)2-s + (−0.385 − 0.510i)3-s + (1.90 − 1.02i)4-s + (−1.22 + 1.42i)5-s + (−0.884 − 0.712i)6-s + (−0.691 + 0.220i)7-s + (1.52 − 1.38i)8-s + (0.161 − 0.568i)9-s + (−1.48 + 2.98i)10-s + (0.254 + 0.262i)11-s + (−1.25 − 0.576i)12-s + (−0.763 − 0.695i)13-s + (−1.09 + 0.678i)14-s + (1.19 + 0.0739i)15-s + (0.827 − 1.24i)16-s + (0.00367 − 0.00295i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73798 - 0.534041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73798 - 0.534041i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (10.1 - 0.373i)T \) |
| good | 2 | \( 1 + (-2.43 + 0.612i)T + (1.76 - 0.946i)T^{2} \) |
| 3 | \( 1 + (0.667 + 0.883i)T + (-0.820 + 2.88i)T^{2} \) |
| 5 | \( 1 + (2.73 - 3.19i)T + (-0.766 - 4.94i)T^{2} \) |
| 7 | \( 1 + (1.82 - 0.582i)T + (5.71 - 4.04i)T^{2} \) |
| 11 | \( 1 + (-0.844 - 0.870i)T + (-0.338 + 10.9i)T^{2} \) |
| 13 | \( 1 + (2.75 + 2.50i)T + (1.19 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0151 + 0.0121i)T + (3.63 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-7.88 - 0.976i)T + (18.4 + 4.63i)T^{2} \) |
| 23 | \( 1 + (-0.529 - 1.86i)T + (-19.5 + 12.1i)T^{2} \) |
| 29 | \( 1 + (-1.09 - 3.10i)T + (-22.5 + 18.1i)T^{2} \) |
| 31 | \( 1 + (-0.104 - 0.209i)T + (-18.6 + 24.7i)T^{2} \) |
| 37 | \( 1 + (0.556 + 6.00i)T + (-36.3 + 6.79i)T^{2} \) |
| 41 | \( 1 + (5.87 + 6.85i)T + (-6.28 + 40.5i)T^{2} \) |
| 43 | \( 1 + (6.10 - 2.80i)T + (27.9 - 32.6i)T^{2} \) |
| 47 | \( 1 + (-2.42 + 4.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.18 - 9.87i)T + (-36.8 + 38.0i)T^{2} \) |
| 59 | \( 1 + (-9.34 - 2.97i)T + (48.1 + 34.0i)T^{2} \) |
| 61 | \( 1 + (0.543 + 0.210i)T + (45.0 + 41.0i)T^{2} \) |
| 67 | \( 1 + (0.241 + 1.10i)T + (-60.8 + 28.0i)T^{2} \) |
| 71 | \( 1 + (1.61 - 4.57i)T + (-55.3 - 44.5i)T^{2} \) |
| 73 | \( 1 + (-6.85 + 1.28i)T + (68.0 - 26.3i)T^{2} \) |
| 79 | \( 1 + (-0.476 - 0.0891i)T + (73.6 + 28.5i)T^{2} \) |
| 83 | \( 1 + (1.50 - 6.88i)T + (-75.4 - 34.6i)T^{2} \) |
| 89 | \( 1 + (-4.54 + 2.81i)T + (39.6 - 79.6i)T^{2} \) |
| 97 | \( 1 + (6.15 + 4.94i)T + (20.7 + 94.7i)T^{2} \) |
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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
−13.67439143112181345204256356301, −12.32224671039689657980178944866, −12.06245815054589511162903412336, −11.12775679888840762603488834440, −9.980179524750747886957096046392, −7.34394478851828412536436712219, −6.80932511065198871838089055546, −5.54572379106323229240438048427, −3.75395513427702369906963696590, −2.96906771792175535141971430032,
3.54892708264887466618449373844, 4.63708430997921350119043256046, 5.23731273980005863079613568878, 6.91203181514017252571781387043, 8.033178195045543648232636406316, 9.681131549772978379124622228290, 11.58646144274627201350603183571, 11.86214328537553598972019548840, 13.01214568853244538086081264541, 13.69058396657894415580829007821
