L(s) = 1 | + (1.37 + 0.331i)2-s + (1.78 + 0.910i)4-s − 2i·5-s + 3.02·7-s + (2.14 + 1.84i)8-s + (0.662 − 2.74i)10-s + (−2.33 + 2.35i)11-s + 1.32·13-s + (4.15 + i)14-s + (2.34 + 3.24i)16-s − 1.69i·17-s − 7.04i·19-s + (1.82 − 3.56i)20-s + (−3.98 + 2.47i)22-s + 3.12i·23-s + ⋯ |
L(s) = 1 | + (0.972 + 0.234i)2-s + (0.890 + 0.455i)4-s − 0.894i·5-s + 1.14·7-s + (0.759 + 0.650i)8-s + (0.209 − 0.869i)10-s + (−0.703 + 0.711i)11-s + 0.367·13-s + (1.10 + 0.267i)14-s + (0.585 + 0.810i)16-s − 0.411i·17-s − 1.61i·19-s + (0.407 − 0.796i)20-s + (−0.850 + 0.526i)22-s + 0.651i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.14169 + 0.129031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.14169 + 0.129031i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.331i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (2.33 - 2.35i)T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 1.69iT - 17T^{2} \) |
| 19 | \( 1 + 7.04iT - 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 1.54T + 29T^{2} \) |
| 31 | \( 1 - 8.30iT - 31T^{2} \) |
| 37 | \( 1 + 4.66iT - 37T^{2} \) |
| 41 | \( 1 + 7.73iT - 41T^{2} \) |
| 43 | \( 1 - 3.95iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 8.24iT - 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 3.08iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 4.71iT - 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−10.68071398710248065479308936809, −9.253126612546877631552566178494, −8.453137990958174892566968660589, −7.57405446695000088378755394455, −6.87096843545751126740453408203, −5.39765890051931408357030446313, −5.00688256695540489659872578706, −4.22889891544584674515297565017, −2.75725373766453698303064917805, −1.51767190767838413111494390601,
1.62443288371988846463433342610, 2.80378986325135388681859307850, 3.79035643900365941917606926398, 4.83339372262381761708286184028, 5.84866369597369114585589786579, 6.47274699331840018462593006905, 7.77755782374693201615312479876, 8.158752157140861115945875236137, 9.765921307738481915548347036755, 10.77821171167380559894466993454