L(s) = 1 | − 1.73i·3-s − 4.29·5-s − 2.75i·7-s − 2.99·9-s + 13.7i·11-s − 14.5·13-s + 7.43i·15-s + 22.8·17-s + 16.0i·19-s − 4.77·21-s + 17.1i·23-s − 6.57·25-s + 5.19i·27-s + 21.8·29-s + 38.6i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.858·5-s − 0.393i·7-s − 0.333·9-s + 1.25i·11-s − 1.11·13-s + 0.495i·15-s + 1.34·17-s + 0.844i·19-s − 0.227·21-s + 0.743i·23-s − 0.262·25-s + 0.192i·27-s + 0.754·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.529413 + 0.529413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.529413 + 0.529413i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
good | 5 | \( 1 + 4.29T + 25T^{2} \) |
| 7 | \( 1 + 2.75iT - 49T^{2} \) |
| 11 | \( 1 - 13.7iT - 121T^{2} \) |
| 13 | \( 1 + 14.5T + 169T^{2} \) |
| 17 | \( 1 - 22.8T + 289T^{2} \) |
| 19 | \( 1 - 16.0iT - 361T^{2} \) |
| 23 | \( 1 - 17.1iT - 529T^{2} \) |
| 29 | \( 1 - 21.8T + 841T^{2} \) |
| 31 | \( 1 - 38.6iT - 961T^{2} \) |
| 37 | \( 1 + 66.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 23.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 14.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 65.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 65.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 74.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 122. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 122.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.3T + 9.40e3T^{2} \) |
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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
−11.65139785645475308532014413332, −10.31881818890008902789779242623, −9.704777955352381772683983520406, −8.285630949424002549973506114337, −7.48683988543571954501379001037, −7.01022146554197453419130685034, −5.49847563534725568695769592865, −4.39224442054104935801887489634, −3.15795273391272359209945141411, −1.52797980866440134138739861454,
0.33762254544829742289622063099, 2.73781710422507282641429163482, 3.77239627911664169032447860409, 4.95346177231044835112952091348, 5.90524763752171809285178136454, 7.27291589523722248802686504807, 8.196752767566621270160141727265, 9.001356840326585867768580553336, 10.04466751312845874985533196009, 10.90408708989971924444634879701