L(s) = 1 | + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (0.987 − 0.156i)5-s + (0.156 + 0.987i)8-s + (0.951 + 0.309i)10-s + (−0.896 − 1.76i)13-s + (−0.309 + 0.951i)16-s + (−1.87 + 0.297i)17-s + (0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s − 1.97i·26-s + (−1.44 + 1.04i)29-s + (−0.707 + 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 − 0.412i)37-s + ⋯ |
L(s) = 1 | + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (0.987 − 0.156i)5-s + (0.156 + 0.987i)8-s + (0.951 + 0.309i)10-s + (−0.896 − 1.76i)13-s + (−0.309 + 0.951i)16-s + (−1.87 + 0.297i)17-s + (0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s − 1.97i·26-s + (−1.44 + 1.04i)29-s + (−0.707 + 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 − 0.412i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.783483901\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783483901\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.987 + 0.156i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (1.87 - 0.297i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.297 - 0.0966i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-1.16 - 0.183i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
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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
−10.62468654645593219656399147934, −9.505827637465213051821098236912, −8.655757768446618748027378154981, −7.68104740642207546812769486796, −6.84713562938236125333663758988, −5.89526077755966659292296768195, −5.27942561312977249183062576581, −4.36469933681150109980861661041, −3.01163886995903842155519242580, −2.10943664465414073280822298894,
1.94411884950686831028261501544, 2.48451853843052787941143155655, 4.08292346278167262587104018299, 4.76275111215434283344644030924, 5.82409855472728154804306542907, 6.65431196213229251477929329147, 7.22408626738866596027990801597, 8.961132016459778387106406467879, 9.495164798497544813885698392455, 10.26200786481254251915715518334