| L(s) = 1 | + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s − 7i·7-s − 8i·8-s − 9·9-s + 12·11-s + 12i·12-s − 58i·13-s + 14·14-s + 16·16-s − 42i·17-s − 18i·18-s + 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.328·11-s + 0.288i·12-s − 1.23i·13-s + 0.267·14-s + 0.250·16-s − 0.599i·17-s − 0.235i·18-s + 0.0482·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5493396483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5493396483\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 11 | \( 1 - 12T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 42iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 294T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 58iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 282T + 6.89e4T^{2} \) |
| 43 | \( 1 - 428iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 384iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 138iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 468T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 556iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 624T + 3.57e5T^{2} \) |
| 73 | \( 1 + 958iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 632T + 4.93e5T^{2} \) |
| 83 | \( 1 - 84iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 - 790iT - 9.12e5T^{2} \) |
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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
−9.093533278509021521441529574386, −8.016717943606218950892259883862, −7.59183670263404014850702540769, −6.71237840203195955975976998321, −5.85576150230475189181022391403, −5.06946145017194583285463051319, −3.88814127812939366455154537007, −2.80584335522772075969002016885, −1.26443006075169980714028359735, −0.14114510459074041409880762665,
1.52401822354506385604392825784, 2.54902564008732746161978721934, 3.78000739551581455468268342337, 4.37184734646256103972229165391, 5.47326297373270929803648566710, 6.34381083988729735734168297521, 7.50191579547575030130936622801, 8.590564842011763680296557651552, 9.242736801170305770586768009112, 9.772056103996564966692998564641
