L(s) = 1 | + (1.22 − 1.22i)3-s + (3.71 − 3.71i)5-s + 5.63·7-s − 2.99i·9-s + (5.26 + 5.26i)11-s + (1.74 + 1.74i)13-s − 9.11i·15-s − 1.43·17-s + (22.1 − 22.1i)19-s + (6.90 − 6.90i)21-s + 2.27·23-s − 2.67i·25-s + (−3.67 − 3.67i)27-s + (35.9 + 35.9i)29-s + 13.3i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.743 − 0.743i)5-s + 0.805·7-s − 0.333i·9-s + (0.478 + 0.478i)11-s + (0.133 + 0.133i)13-s − 0.607i·15-s − 0.0846·17-s + (1.16 − 1.16i)19-s + (0.328 − 0.328i)21-s + 0.0988·23-s − 0.106i·25-s + (−0.136 − 0.136i)27-s + (1.23 + 1.23i)29-s + 0.429i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.870998646\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.870998646\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-3.71 + 3.71i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.63T + 49T^{2} \) |
| 11 | \( 1 + (-5.26 - 5.26i)T + 121iT^{2} \) |
| 13 | \( 1 + (-1.74 - 1.74i)T + 169iT^{2} \) |
| 17 | \( 1 + 1.43T + 289T^{2} \) |
| 19 | \( 1 + (-22.1 + 22.1i)T - 361iT^{2} \) |
| 23 | \( 1 - 2.27T + 529T^{2} \) |
| 29 | \( 1 + (-35.9 - 35.9i)T + 841iT^{2} \) |
| 31 | \( 1 - 13.3iT - 961T^{2} \) |
| 37 | \( 1 + (36.1 - 36.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 63.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (46.5 + 46.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 61.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (49.0 - 49.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-41.6 - 41.6i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-40.4 - 40.4i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-36.0 + 36.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 8.24T + 5.04e3T^{2} \) |
| 73 | \( 1 + 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 93.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-21.6 + 21.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115.T + 9.40e3T^{2} \) |
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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
−9.847460794335858516649307483587, −8.909001795800386475536053912110, −8.591597273750008613758591429490, −7.32060054650216390676765226365, −6.68680211851307397793556606540, −5.29513165386547111030526605548, −4.81156914026086869164089298148, −3.34048780149113464012386845377, −1.95755347098378130782062651036, −1.11658000219937652440531278062,
1.42531927833009135110935315193, 2.65597777391784335153341063982, 3.66109019463509470509734907315, 4.83124188993843256059122940562, 5.85687518785634387519904028455, 6.65046083102533041019700514925, 7.912540517364479797344286596038, 8.400015573703033601052732369726, 9.797365869666674298853337899517, 9.892589460067679811612323171978