| L(s) = 1 | + (−1.18 − 0.429i)2-s + (0.523 + 2.96i)3-s + (−0.321 − 0.270i)4-s + (0.766 − 0.642i)5-s + (0.657 − 3.72i)6-s + (−1.86 + 3.22i)7-s + (1.52 + 2.63i)8-s + (−5.70 + 2.07i)9-s + (−1.18 + 0.429i)10-s + (1.67 + 2.90i)11-s + (0.632 − 1.09i)12-s + (0.840 − 4.76i)13-s + (3.58 − 3.00i)14-s + (2.30 + 1.93i)15-s + (−0.518 − 2.93i)16-s + (2.51 + 0.914i)17-s + ⋯ |
| L(s) = 1 | + (−0.835 − 0.303i)2-s + (0.301 + 1.71i)3-s + (−0.160 − 0.135i)4-s + (0.342 − 0.287i)5-s + (0.268 − 1.52i)6-s + (−0.703 + 1.21i)7-s + (0.537 + 0.931i)8-s + (−1.90 + 0.692i)9-s + (−0.373 + 0.135i)10-s + (0.505 + 0.876i)11-s + (0.182 − 0.316i)12-s + (0.233 − 1.32i)13-s + (0.958 − 0.804i)14-s + (0.595 + 0.499i)15-s + (−0.129 − 0.734i)16-s + (0.609 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0868 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0868 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.487501 + 0.446864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487501 + 0.446864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.961 + 4.25i)T \) |
| good | 2 | \( 1 + (1.18 + 0.429i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.523 - 2.96i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (1.86 - 3.22i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.840 + 4.76i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.51 - 0.914i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.43 - 1.20i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.93 + 1.79i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 2.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.992T + 37T^{2} \) |
| 41 | \( 1 + (-0.0723 - 0.410i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.52 - 4.64i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (2.10 - 0.766i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.199 - 0.167i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.87 - 1.77i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.589 - 0.494i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (10.1 - 3.68i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 1.28i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.792 - 4.49i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.09 + 11.8i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.78 + 11.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.33 - 7.55i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.79 - 2.47i)T + (74.3 + 62.3i)T^{2} \) |
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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
−14.68535478187445827698721871832, −13.22976113108519329489014357271, −11.78827936965397452400423718471, −10.47476201103686186902460199161, −9.745428776403270188647732670116, −9.197833044386932484227401333224, −8.292173175003709750612888291177, −5.73444294779860293231109864750, −4.74793906638631471581656785927, −2.88802175680079804004633903380,
1.15578292443135587314897622014, 3.54062394375671546446188587085, 6.46189110692043399339774373079, 6.96537460116474785670218055334, 8.034633160060860581164044440035, 9.051715338483199949914630650904, 10.28046597969541994469412194214, 11.81106401054196548142316944405, 12.90539527420452287120436330927, 13.90034684582859598556948156631
