| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.23i·5-s − 7.05i·7-s + 2.82·8-s − 3.16i·10-s + 10.4i·11-s + 19.0·13-s − 9.98i·14-s + 4.00·16-s + 12.8i·17-s + 22.7i·19-s − 4.47i·20-s + 14.7i·22-s + (6.14 − 22.1i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.447i·5-s − 1.00i·7-s + 0.353·8-s − 0.316i·10-s + 0.950i·11-s + 1.46·13-s − 0.713i·14-s + 0.250·16-s + 0.754i·17-s + 1.19i·19-s − 0.223i·20-s + 0.671i·22-s + (0.266 − 0.963i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.724206294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.724206294\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-6.14 + 22.1i)T \) |
| good | 7 | \( 1 + 7.05iT - 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 - 12.8iT - 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 29 | \( 1 - 8.61T + 841T^{2} \) |
| 31 | \( 1 - 22.2T + 961T^{2} \) |
| 37 | \( 1 - 29.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 10.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 74.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 122.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 140. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 11.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 57.0iT - 9.40e3T^{2} \) |
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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
−8.680461504062365466052109326168, −8.135917831498670017521480563597, −7.22762895940948914140326511737, −6.43476462233512290767344791207, −5.77691835384670370841389038559, −4.61040164362419473914394403407, −4.12768236141921785163686499283, −3.31129755654889917306088032480, −1.87901989536278882770142341912, −0.983160171761048624828664274312,
0.954199757479413208246783811648, 2.40233420743063437181120549287, 3.10353252527293129778952876189, 3.93513394448119844476873094234, 5.12588133145927395572400268163, 5.76252856351137717456389263386, 6.43043425383420626622276221357, 7.21703558916306151115743243605, 8.287577020206282637792224594748, 8.873732725559643135724434451587
