| L(s) = 1 | + (−13.2 + 13.2i)3-s + (25.6 + 25.6i)7-s − 109. i·9-s + 16.4i·11-s + (−212. − 212. i)13-s + (−642. + 642. i)17-s + 1.28e3·19-s − 680.·21-s + (562. − 562. i)23-s + (−1.77e3 − 1.77e3i)27-s − 6.97e3i·29-s − 5.01e3i·31-s + (−218. − 218. i)33-s + (−1.05e4 + 1.05e4i)37-s + 5.65e3·39-s + ⋯ |
| L(s) = 1 | + (−0.851 + 0.851i)3-s + (0.197 + 0.197i)7-s − 0.450i·9-s + 0.0410i·11-s + (−0.349 − 0.349i)13-s + (−0.538 + 0.538i)17-s + 0.817·19-s − 0.336·21-s + (0.221 − 0.221i)23-s + (−0.468 − 0.468i)27-s − 1.54i·29-s − 0.937i·31-s + (−0.0349 − 0.0349i)33-s + (−1.26 + 1.26i)37-s + 0.595·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.148561092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148561092\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (13.2 - 13.2i)T - 243iT^{2} \) |
| 7 | \( 1 + (-25.6 - 25.6i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 16.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (212. + 212. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (642. - 642. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-562. + 562. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 6.97e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.01e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.05e4 - 1.05e4i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-5.19e3 + 5.19e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-42.6 - 42.6i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.51e3 - 1.51e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 2.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.74e4 - 2.74e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 4.57e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-5.31e3 - 5.31e3i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.61e4 + 6.61e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 7.93e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.20e4 + 2.20e4i)T - 8.58e9iT^{2} \) |
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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
−10.33336438466265186985749336880, −9.909602116768117358495134303654, −8.723285782939010342367463828888, −7.71927941011797798551815019819, −6.47640437822046604094480536567, −5.48689675816424520499442670821, −4.75878510436976975525261031607, −3.68913689550656851295028078828, −2.17463375138466374352454576190, −0.45926470108804124668582994248,
0.76032977420175917666811623768, 1.82064269418928209743482955973, 3.36269600396745704261432872328, 4.86707612530125589725880613719, 5.66310785400612599589560337348, 6.96856188117391888023914610433, 7.17339669677506096804203818056, 8.585194596065532048298470459127, 9.531559127953733756771628787425, 10.72462471285561446280812449350
