| L(s) = 1 | + (1.57 + 0.719i)3-s + (−2.68 − 2.32i)5-s + (−0.314 + 0.202i)7-s + (−0.00255 − 0.00294i)9-s + (−3.35 − 0.482i)11-s + (−0.0151 + 0.0234i)13-s + (−2.55 − 5.58i)15-s + (−0.162 − 0.0478i)17-s + (−1.16 − 3.97i)19-s + (−0.641 + 0.0921i)21-s + (−4.32 − 2.07i)23-s + (1.07 + 7.50i)25-s + (−1.46 − 4.98i)27-s + (2.21 − 7.54i)29-s + (1.41 + 3.10i)31-s + ⋯ |
| L(s) = 1 | + (0.909 + 0.415i)3-s + (−1.19 − 1.03i)5-s + (−0.118 + 0.0764i)7-s + (−0.000852 − 0.000983i)9-s + (−1.01 − 0.145i)11-s + (−0.00418 + 0.00651i)13-s + (−0.658 − 1.44i)15-s + (−0.0395 − 0.0115i)17-s + (−0.267 − 0.912i)19-s + (−0.139 + 0.0201i)21-s + (−0.901 − 0.431i)23-s + (0.215 + 1.50i)25-s + (−0.281 − 0.960i)27-s + (0.411 − 1.40i)29-s + (0.254 + 0.557i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.355810 - 0.729134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355810 - 0.729134i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (4.32 + 2.07i)T \) |
| good | 3 | \( 1 + (-1.57 - 0.719i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (2.68 + 2.32i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.314 - 0.202i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (3.35 + 0.482i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.0151 - 0.0234i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.162 + 0.0478i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.16 + 3.97i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.21 + 7.54i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.41 - 3.10i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (8.60 - 7.45i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.85 + 5.60i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (9.01 + 4.11i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 + (0.748 + 1.16i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.06 - 1.66i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-7.17 + 3.27i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-7.10 + 1.02i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.00700 - 0.0486i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 3.42i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.97 - 3.84i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.52 + 3.91i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.81 - 14.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (3.32 - 3.84i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.912324438040827468840659988268, −9.027380827376708990887111126213, −8.242990990279389203817104336208, −8.020083058095200735994506473929, −6.69370469857103227807431448089, −5.28712802300875171840963466953, −4.41591177774351203348407605211, −3.58069570023389419973871144667, −2.47306103122792180644587775915, −0.35083811701692523310922597213,
2.13301189566556734249775105613, 3.15718180424057903001405376910, 3.85758148303048485216124664809, 5.28610634506472027429170959123, 6.61181533825554472591152031841, 7.48547929775077379497568379392, 7.952592172907131319210805101063, 8.642236620362547289370693720448, 9.958678994273838496134164248915, 10.68393819181824507257148199583
