L(s) = 1 | + 0.747i·2-s + (4.77 + 2.05i)3-s + 7.44·4-s + (4.35 − 7.53i)5-s + (−1.53 + 3.56i)6-s + (−11.6 − 14.4i)7-s + 11.5i·8-s + (18.5 + 19.5i)9-s + (5.63 + 3.25i)10-s + (−35.9 + 20.7i)11-s + (35.5 + 15.2i)12-s + (33.0 − 19.0i)13-s + (10.7 − 8.68i)14-s + (36.2 − 27.0i)15-s + 50.8·16-s + (−39.9 + 69.2i)17-s + ⋯ |
L(s) = 1 | + 0.264i·2-s + (0.918 + 0.394i)3-s + 0.930·4-s + (0.389 − 0.674i)5-s + (−0.104 + 0.242i)6-s + (−0.627 − 0.778i)7-s + 0.510i·8-s + (0.688 + 0.725i)9-s + (0.178 + 0.102i)10-s + (−0.985 + 0.568i)11-s + (0.854 + 0.367i)12-s + (0.704 − 0.406i)13-s + (0.205 − 0.165i)14-s + (0.623 − 0.465i)15-s + 0.795·16-s + (−0.570 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.12588 + 0.367238i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12588 + 0.367238i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.77 - 2.05i)T \) |
| 7 | \( 1 + (11.6 + 14.4i)T \) |
good | 2 | \( 1 - 0.747iT - 8T^{2} \) |
| 5 | \( 1 + (-4.35 + 7.53i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (35.9 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-33.0 + 19.0i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (39.9 - 69.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (49.4 - 28.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (153. + 88.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (60.1 + 34.7i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 260. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (74.9 + 129. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-54.7 - 94.7i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-124. + 215. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 295.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (263. + 152. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 720.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 648. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 194.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 98.7iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-226. - 130. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-285. + 494. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (89.5 + 155. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-956. - 552. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.71755557235785848198812086402, −13.34593049727106486747447408616, −12.70942946625403138177628832012, −10.72743022402654889468903777905, −10.03376846911420127490967607239, −8.497276174989016092249596230283, −7.48485538080292167527194057722, −5.94048042591268153725138479919, −4.02765286990039475001600478699, −2.17403201635985753344483416767,
2.23591213537052486286734000014, 3.20147163017417786354266249979, 6.09031030762297656745562851620, 7.06452279296784414590664419714, 8.519148932329782574188056942370, 9.824714596740955420771800559018, 10.98748040751398559921084011772, 12.24433450836725649509136507180, 13.33837847554443294342554928935, 14.30413831042165842458701708498