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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.670154 - 1.13579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.670154 - 1.13579i\) |
\(L(\frac{5}{2})\) |
|
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
−12.67188366051350463558344247175, −11.85384250301024020452756145407, −10.40022648199925015166587646180, −9.263315272965089253733276187560, −8.448200868229780336654604260242, −6.73408721364126725352949659270, −5.76946461758849453315252436969, −4.86978075734467115733116009140, −2.33616594402393225261548386729, −0.72746705590089028039057230126,
2.13572833002157038585733130948, 3.97749720127970560908949189925, 5.43766164483573018685714363875, 6.47574726169490169260452060889, 7.58804450765820190788745155088, 9.467403888782344626711385241688, 10.31770807799437992602459024464, 10.76245701456764122678890190076, 11.97758683516929906352774586758, 13.48845681870954573521813068359