| L(s) = 1 | + (0.909 + 1.08i)2-s + (2.14 − 2.09i)3-s + (−0.347 + 1.96i)4-s + (0.387 + 1.06i)5-s + (4.22 + 0.418i)6-s + (−0.332 − 1.88i)7-s + (−2.44 + 1.41i)8-s + (0.205 − 8.99i)9-s + (−0.800 + 1.38i)10-s + (−6.05 + 16.6i)11-s + (3.38 + 4.95i)12-s + (−9.14 − 7.67i)13-s + (1.73 − 2.07i)14-s + (3.06 + 1.47i)15-s + (−3.75 − 1.36i)16-s + (−2.93 − 1.69i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (0.715 − 0.698i)3-s + (−0.0868 + 0.492i)4-s + (0.0774 + 0.212i)5-s + (0.703 + 0.0696i)6-s + (−0.0474 − 0.269i)7-s + (−0.306 + 0.176i)8-s + (0.0228 − 0.999i)9-s + (−0.0800 + 0.138i)10-s + (−0.550 + 1.51i)11-s + (0.282 + 0.412i)12-s + (−0.703 − 0.590i)13-s + (0.124 − 0.148i)14-s + (0.204 + 0.0980i)15-s + (−0.234 − 0.0855i)16-s + (−0.172 − 0.0998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61130 + 0.231750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61130 + 0.231750i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 3 | \( 1 + (-2.14 + 2.09i)T \) |
| good | 5 | \( 1 + (-0.387 - 1.06i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (0.332 + 1.88i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (6.05 - 16.6i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (9.14 + 7.67i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.93 + 1.69i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (10.2 + 17.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.678 - 0.119i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-34.6 - 41.3i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (7.33 - 41.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-8.24 + 14.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-26.0 + 30.9i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-66.4 - 24.1i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (62.9 - 11.1i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 17.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (19.1 + 52.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 65.3i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (17.2 + 14.5i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (43.0 + 24.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (45.2 + 78.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (57.2 - 48.0i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-8.15 - 9.72i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-58.5 + 33.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (44.2 + 16.1i)T + (7.20e3 + 6.04e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.90772595066502435647568610329, −14.22230990021033691002408344258, −12.88226591393590831507570933532, −12.39250973225327104725136708672, −10.41008307445508280843543008784, −8.926589304882862890056795410043, −7.51851863961740118387786042875, −6.76710677094160569160493727486, −4.75726040545208337705403596433, −2.69634484913065404676123428739,
2.68933398913361649537474341982, 4.28662529850780581039586903138, 5.81215819023315919655327727023, 8.103107543721627101831930090124, 9.258061038737906162539983676691, 10.42199727332834172671937324208, 11.53409530170406675839759242730, 13.00294870510459137150550750266, 13.93630217019487460819943914325, 14.87122689168945020002994630760
