| L(s) = 1 | + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (−6.84 − 1.46i)7-s + 2.82·8-s − 2.99·9-s − 5.87·11-s + 3.46i·12-s − 2.06i·13-s + (−9.68 − 2.07i)14-s + 4.00·16-s − 2.59i·17-s − 4.24·18-s − 14.3i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (−0.977 − 0.209i)7-s + 0.353·8-s − 0.333·9-s − 0.534·11-s + 0.288i·12-s − 0.158i·13-s + (−0.691 − 0.148i)14-s + 0.250·16-s − 0.152i·17-s − 0.235·18-s − 0.757i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.219834487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219834487\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.84 + 1.46i)T \) |
| good | 11 | \( 1 + 5.87T + 121T^{2} \) |
| 13 | \( 1 + 2.06iT - 169T^{2} \) |
| 17 | \( 1 + 2.59iT - 289T^{2} \) |
| 19 | \( 1 + 14.3iT - 361T^{2} \) |
| 23 | \( 1 - 4.73T + 529T^{2} \) |
| 29 | \( 1 + 16.1T + 841T^{2} \) |
| 31 | \( 1 + 48.3iT - 961T^{2} \) |
| 37 | \( 1 + 46.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 74.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 73.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 39.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 11.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 41.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 92.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 4.26iT - 9.40e3T^{2} \) |
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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
−9.581738403366987886561235386766, −8.823392944850176609602694184142, −7.62406852769710638192018681985, −6.86210460145915421956824080282, −5.87980940107992504058076195077, −5.15021799844557569541621881113, −4.09649557402825312328630219235, −3.30171565134772201264409684474, −2.32141481013007993713759164870, −0.27606560473243984537691192838,
1.52868223311098202869135846693, 2.80155206148151853229010811896, 3.54547673458411955860458430087, 4.83371692399453286033498889162, 5.78152580303200286137542241467, 6.48229985059725037709073990327, 7.25419470520618109355774618655, 8.152585041539815025088027093727, 9.104231833498079147209992640280, 10.07681176226162176159034237587
