L(s) = 1 | + (3.27 + 3.27i)3-s + (−12.6 + 12.6i)5-s + 13.8i·7-s − 5.59i·9-s + (−1.54 + 1.54i)11-s + (32.7 + 32.7i)13-s − 82.7·15-s + 18.6·17-s + (86.4 + 86.4i)19-s + (−45.3 + 45.3i)21-s − 134. i·23-s − 194. i·25-s + (106. − 106. i)27-s + (−59.7 − 59.7i)29-s + 31.5·31-s + ⋯ |
L(s) = 1 | + (0.629 + 0.629i)3-s + (−1.13 + 1.13i)5-s + 0.749i·7-s − 0.207i·9-s + (−0.0424 + 0.0424i)11-s + (0.699 + 0.699i)13-s − 1.42·15-s + 0.266·17-s + (1.04 + 1.04i)19-s + (−0.471 + 0.471i)21-s − 1.21i·23-s − 1.55i·25-s + (0.760 − 0.760i)27-s + (−0.382 − 0.382i)29-s + 0.182·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.902146 + 1.06883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.902146 + 1.06883i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-3.27 - 3.27i)T + 27iT^{2} \) |
| 5 | \( 1 + (12.6 - 12.6i)T - 125iT^{2} \) |
| 7 | \( 1 - 13.8iT - 343T^{2} \) |
| 11 | \( 1 + (1.54 - 1.54i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-32.7 - 32.7i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 18.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-86.4 - 86.4i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 134. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (59.7 + 59.7i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-89.1 + 89.1i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 210. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (119. - 119. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (26.1 - 26.1i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (441. - 441. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (174. + 174. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (91.7 + 91.7i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 299. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 943.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (313. + 313. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−14.82383038589923523958137698387, −14.09906715063862495820388102122, −12.23628056707127997570096392037, −11.38906105009746692706596602011, −10.14476444848607204978828419390, −8.888645356989188218768884474093, −7.73989611752257137067065004240, −6.27636006589347339299812053509, −4.08608695651972294645495602769, −2.99285370042561558265437297371,
0.993781933026243060606082500064, 3.52383567185329220876673831116, 5.09365977207373025911292522983, 7.35745855625804172472271602582, 8.009631817971933815054370164289, 9.123115032198351272194554372840, 10.87925272843315682559087466588, 12.05027251258514323618281047279, 13.18979160537757419926045199101, 13.75278204691826082885822742695