| 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
−13.36713239730757682843095593640, −12.58093036157324492483775612171, −11.33939064989649919578646807571, −9.695549365076392503002088535454, −9.010484205403930875781914594549, −7.70122331784626184246561586672, −6.90100728763865875786995383329, −5.40410256602741021449250087063, −4.40942531140105547465384906918, −1.87233568129867072492295421085,
2.32714483661903907404796845019, 3.59839527451307880607111903366, 5.25486964140123537439937979256, 6.36551773853190564353564352570, 8.302981658484920195089875631105, 9.056063483119730713962239800975, 10.48196516847350860961507709473, 10.84957581993987423501264403715, 11.91075791300733914431725357184, 13.51873932555049349241231201169
