| 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 | \( 1 + (44.2 - 16.1i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−14.87122689168945020002994630760, −13.93630217019487460819943914325, −13.00294870510459137150550750266, −11.53409530170406675839759242730, −10.42199727332834172671937324208, −9.258061038737906162539983676691, −8.103107543721627101831930090124, −5.81215819023315919655327727023, −4.28662529850780581039586903138, −2.68933398913361649537474341982,
2.69634484913065404676123428739, 4.75726040545208337705403596433, 6.76710677094160569160493727486, 7.51851863961740118387786042875, 8.926589304882862890056795410043, 10.41008307445508280843543008784, 12.39250973225327104725136708672, 12.88226591393590831507570933532, 14.22230990021033691002408344258, 14.90772595066502435647568610329
