| L(s) = 1 | + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (1.59 + 1.83i)5-s + (0.959 − 0.281i)6-s + (1.56 + 1.00i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (2.04 − 1.31i)10-s + (−0.790 − 5.49i)11-s + (−0.142 − 0.989i)12-s + (0.966 − 0.620i)13-s + (1.21 − 1.40i)14-s + (−1.01 + 2.21i)15-s + (0.841 + 0.540i)16-s + (−7.20 + 2.11i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.712 + 0.822i)5-s + (0.391 − 0.115i)6-s + (0.590 + 0.379i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.647 − 0.415i)10-s + (−0.238 − 1.65i)11-s + (−0.0410 − 0.285i)12-s + (0.267 − 0.172i)13-s + (0.324 − 0.374i)14-s + (−0.260 + 0.571i)15-s + (0.210 + 0.135i)16-s + (−1.74 + 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29224 - 0.0911314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29224 - 0.0911314i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.45 + 4.12i)T \) |
| good | 5 | \( 1 + (-1.59 - 1.83i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.56 - 1.00i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.790 + 5.49i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.966 + 0.620i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (7.20 - 2.11i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.29 - 1.26i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.81 - 0.827i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.863 - 1.88i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.06 - 5.84i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.450 + 0.520i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.55 + 7.79i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 + (-4.53 - 2.91i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.39 - 2.18i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (0.157 - 0.344i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.420 - 2.92i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-2.14 + 14.9i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 3.23i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-8.86 + 5.69i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (6.84 - 7.89i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.47 - 9.80i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.51 - 7.51i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.51873932555049349241231201169, −11.91075791300733914431725357184, −10.84957581993987423501264403715, −10.48196516847350860961507709473, −9.056063483119730713962239800975, −8.302981658484920195089875631105, −6.36551773853190564353564352570, −5.25486964140123537439937979256, −3.59839527451307880607111903366, −2.32714483661903907404796845019,
1.87233568129867072492295421085, 4.40942531140105547465384906918, 5.40410256602741021449250087063, 6.90100728763865875786995383329, 7.70122331784626184246561586672, 9.010484205403930875781914594549, 9.695549365076392503002088535454, 11.33939064989649919578646807571, 12.58093036157324492483775612171, 13.36713239730757682843095593640
