| L(s) = 1 | + (2.86 + 4.09i)5-s + 7.15i·7-s + 5.06i·11-s + 3.12i·13-s − 8.72·17-s − 20.1·19-s + 14.7·23-s + (−8.59 + 23.4i)25-s − 39.7i·29-s − 39.3·31-s + (−29.3 + 20.4i)35-s + 34.8i·37-s − 13.2i·41-s + 66.6i·43-s + 16.9·47-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.819i)5-s + 1.02i·7-s + 0.460i·11-s + 0.240i·13-s − 0.513·17-s − 1.06·19-s + 0.640·23-s + (−0.343 + 0.939i)25-s − 1.37i·29-s − 1.27·31-s + (−0.837 + 0.585i)35-s + 0.942i·37-s − 0.323i·41-s + 1.54i·43-s + 0.359·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.457289469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457289469\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.86 - 4.09i)T \) |
| good | 7 | \( 1 - 7.15iT - 49T^{2} \) |
| 11 | \( 1 - 5.06iT - 121T^{2} \) |
| 13 | \( 1 - 3.12iT - 169T^{2} \) |
| 17 | \( 1 + 8.72T + 289T^{2} \) |
| 19 | \( 1 + 20.1T + 361T^{2} \) |
| 23 | \( 1 - 14.7T + 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 39.3T + 961T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 66.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.62T + 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 106. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 23.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 66.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 175. iT - 9.40e3T^{2} \) |
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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
−10.96600278665774747828127820072, −9.970027491227189345587363116874, −9.244612397095696272619797391100, −8.359113336465934368853472919984, −7.14965074264055476362853282340, −6.33605370284512729798904460401, −5.51443902568099902141323608903, −4.26250676419046964951487041762, −2.79429472960029197822341603246, −1.95837056112683960621635953541,
0.54311710254509418368073126319, 1.92196306624180985342596641615, 3.57492943798616416539456564058, 4.62564200572480354485948660772, 5.58384010389784811522242015628, 6.65695917651390382627637725198, 7.59480249378731988578623493736, 8.741159264459375403060745522948, 9.231231846644878248806631888501, 10.60150862251174203174400637359
