| L(s) = 1 | + (0.873 + 0.185i)2-s + (0.508 + 1.56i)3-s + (−1.09 − 0.489i)4-s + (−0.0637 − 0.606i)5-s + (0.153 + 1.46i)6-s + (−0.455 − 0.506i)7-s + (−2.31 − 1.68i)8-s + (0.236 − 0.171i)9-s + (0.0569 − 0.541i)10-s − 0.856·11-s + (0.206 − 1.96i)12-s + (−0.256 + 0.444i)13-s + (−0.304 − 0.526i)14-s + (0.917 − 0.408i)15-s + (−0.0986 − 0.109i)16-s + (0.929 + 0.413i)17-s + ⋯ |
| L(s) = 1 | + (0.617 + 0.131i)2-s + (0.293 + 0.903i)3-s + (−0.549 − 0.244i)4-s + (−0.0285 − 0.271i)5-s + (0.0626 + 0.596i)6-s + (−0.172 − 0.191i)7-s + (−0.817 − 0.594i)8-s + (0.0787 − 0.0572i)9-s + (0.0180 − 0.171i)10-s − 0.258·11-s + (0.0597 − 0.568i)12-s + (−0.0711 + 0.123i)13-s + (−0.0812 − 0.140i)14-s + (0.236 − 0.105i)15-s + (−0.0246 − 0.0273i)16-s + (0.225 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04164 + 0.258634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04164 + 0.258634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (2.66 + 7.34i)T \) |
| good | 2 | \( 1 + (-0.873 - 0.185i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.508 - 1.56i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.0637 + 0.606i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (0.455 + 0.506i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + 0.856T + 11T^{2} \) |
| 13 | \( 1 + (0.256 - 0.444i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.929 - 0.413i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (2.82 - 3.14i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (6.80 - 4.94i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.37 + 5.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-10.6 + 2.25i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.711 - 2.19i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.10 - 3.41i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.98 + 1.32i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (1.86 + 3.23i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.74 + 3.44i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (8.66 + 1.84i)T + (53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 11.2i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-0.576 + 5.48i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.289 + 2.75i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (6.78 - 3.02i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 2.24i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.60 - 8.02i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.4 + 2.21i)T + (88.6 - 39.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.12517960656133838662436446543, −14.13803937040667304385531639302, −13.15126754316565461511879418725, −12.02014714490073937881091648080, −10.21741443323160488387484815425, −9.581568983529531076236517591079, −8.236352715557703396937381809681, −6.19039090227180915778868104469, −4.72044003635323499202402917757, −3.69750157026722478004108017477,
2.75690566400718835815927404168, 4.63663731144954020387100314420, 6.35778179184274431864767363462, 7.79210869801047866391052143784, 8.900426021970528278016888791573, 10.52997255156455704708677599586, 12.17493552634970634963757062232, 12.78426684639936950356893921103, 13.76530614989847301579732322835, 14.54790875585511088314932456768
