| L(s) = 1 | + (1.38 + 0.280i)2-s + (0.708 + 1.55i)3-s + (1.84 + 0.777i)4-s + (−0.680 − 0.785i)5-s + (0.547 + 2.35i)6-s + (−1.52 − 0.978i)7-s + (2.33 + 1.59i)8-s + (0.0576 − 0.0665i)9-s + (−0.723 − 1.27i)10-s + (−2.78 + 0.400i)11-s + (0.100 + 3.41i)12-s + (−1.06 − 1.66i)13-s + (−1.83 − 1.78i)14-s + (0.736 − 1.61i)15-s + (2.79 + 2.86i)16-s + (−0.326 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.980 + 0.198i)2-s + (0.409 + 0.896i)3-s + (0.921 + 0.388i)4-s + (−0.304 − 0.351i)5-s + (0.223 + 0.959i)6-s + (−0.575 − 0.369i)7-s + (0.826 + 0.563i)8-s + (0.0192 − 0.0221i)9-s + (−0.228 − 0.404i)10-s + (−0.840 + 0.120i)11-s + (0.0289 + 0.984i)12-s + (−0.296 − 0.461i)13-s + (−0.490 − 0.476i)14-s + (0.190 − 0.416i)15-s + (0.698 + 0.715i)16-s + (−0.0791 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88642 + 0.815735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88642 + 0.815735i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.280i)T \) |
| 23 | \( 1 + (2.59 + 4.03i)T \) |
| good | 3 | \( 1 + (-0.708 - 1.55i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.680 + 0.785i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.52 + 0.978i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (2.78 - 0.400i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.06 + 1.66i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.326 + 1.11i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.13 - 3.86i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.543 - 1.85i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.29 - 1.50i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.398 - 0.460i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.94 + 5.70i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.20 + 1.46i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 11.3iT - 47T^{2} \) |
| 53 | \( 1 + (0.454 + 0.291i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.00 - 5.14i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.76 - 12.6i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-7.52 - 1.08i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (8.75 + 1.25i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.33 - 1.85i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.58 + 2.94i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.507 + 0.439i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.31 + 3.33i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (2.21 - 1.92i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.68746342551058698941977384014, −12.16546266763089760138953604052, −10.57632347367974233486578086976, −10.10373157206429395694117012535, −8.606069752877660083279652484367, −7.55436989129750250238448170034, −6.27101941019757998293721432297, −4.89895580500995769452988786077, −4.01695827122730167402398803945, −2.85768548338793568435409339310,
2.12477373663712292870187608992, 3.25137310857604300073228950449, 4.85977636934392215669143391631, 6.25342579384019871431355459386, 7.15292102505018109352277439592, 8.048986123497093239047709164988, 9.651873836241332316512460441307, 10.82392023074312892279633830224, 11.78424895664737818866163155482, 12.73247425878006515826628586757
