| L(s) = 1 | + (−0.877 − 0.154i)2-s + (−0.824 + 1.52i)3-s + (−1.13 − 0.412i)4-s + (−1.30 − 0.474i)5-s + (0.959 − 1.20i)6-s + (1.81 − 1.92i)7-s + (2.47 + 1.42i)8-s + (−1.63 − 2.51i)9-s + (1.07 + 0.617i)10-s + (−0.197 − 0.543i)11-s + (1.56 − 1.38i)12-s + (1.85 − 5.11i)13-s + (−1.88 + 1.41i)14-s + (1.79 − 1.59i)15-s + (−0.102 − 0.0862i)16-s + (1.15 − 2.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 − 0.109i)2-s + (−0.476 + 0.879i)3-s + (−0.566 − 0.206i)4-s + (−0.582 − 0.212i)5-s + (0.391 − 0.493i)6-s + (0.684 − 0.728i)7-s + (0.874 + 0.505i)8-s + (−0.546 − 0.837i)9-s + (0.338 + 0.195i)10-s + (−0.0596 − 0.163i)11-s + (0.451 − 0.400i)12-s + (0.515 − 1.41i)13-s + (−0.504 + 0.377i)14-s + (0.463 − 0.411i)15-s + (−0.0257 − 0.0215i)16-s + (0.280 − 0.485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.422255 - 0.290823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422255 - 0.290823i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.824 - 1.52i)T \) |
| 7 | \( 1 + (-1.81 + 1.92i)T \) |
| good | 2 | \( 1 + (0.877 + 0.154i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (1.30 + 0.474i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.197 + 0.543i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.85 + 5.11i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 2.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.96 + 1.71i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.58 - 1.16i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.224 + 0.616i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.595 + 1.63i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 + (1.37 + 0.500i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.681 - 3.86i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.79 + 2.10i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (8.16 - 4.71i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.88 + 3.25i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.62 - 12.7i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.43 + 8.11i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-13.6 + 7.86i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2.39iT - 73T^{2} \) |
| 79 | \( 1 + (0.465 - 2.64i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 4.10i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.91 + 10.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (18.3 + 3.23i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.04493174883944478887798091723, −11.10490026288231626114137368681, −10.36632076025485722250953150547, −9.593431773234917297890112711452, −8.351447744962157368468085525087, −7.70513056270697087999114479723, −5.72453298082950767211053734353, −4.73905635042380443106800641008, −3.69547933983498564152822290071, −0.64931484658467279525431670251,
1.72383350874307584743511579239, 4.02352900309105837445132607656, 5.42153510929736149323155568210, 6.77492613869746413637777497166, 7.88655827016891422163956023388, 8.428291267842406574280892571741, 9.619865194028743925476972240984, 10.97824595079148287399690919602, 11.83861612603054047135690707453, 12.49130767910349486331962146755
