L(s) = 1 | + (0.316 + 1.97i)2-s + (−3.79 + 1.25i)4-s − 4.41i·5-s − 3.59i·7-s + (−3.67 − 7.10i)8-s + (8.71 − 1.39i)10-s − 18.7·11-s − 19.9i·13-s + (7.10 − 1.14i)14-s + (12.8 − 9.51i)16-s − 8.00·17-s + 16.1·19-s + (5.52 + 16.7i)20-s + (−5.95 − 37.0i)22-s − 18.3i·23-s + ⋯ |
L(s) = 1 | + (0.158 + 0.987i)2-s + (−0.949 + 0.312i)4-s − 0.882i·5-s − 0.514i·7-s + (−0.459 − 0.888i)8-s + (0.871 − 0.139i)10-s − 1.70·11-s − 1.53i·13-s + (0.507 − 0.0814i)14-s + (0.804 − 0.594i)16-s − 0.470·17-s + 0.849·19-s + (0.276 + 0.838i)20-s + (−0.270 − 1.68i)22-s − 0.799i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.764248 - 0.465110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.764248 - 0.465110i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.316 - 1.97i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4.41iT - 25T^{2} \) |
| 7 | \( 1 + 3.59iT - 49T^{2} \) |
| 11 | \( 1 + 18.7T + 121T^{2} \) |
| 13 | \( 1 + 19.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.00T + 289T^{2} \) |
| 19 | \( 1 - 16.1T + 361T^{2} \) |
| 23 | \( 1 + 18.3iT - 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 - 29.7iT - 961T^{2} \) |
| 37 | \( 1 + 26.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 71.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 23.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 90.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 10.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 90.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 74.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 13.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 8.08T + 6.88e3T^{2} \) |
| 89 | \( 1 - 83.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 79.0T + 9.40e3T^{2} \) |
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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
−12.46135999067286149575989110961, −10.71110799553165064580370151964, −9.897195346374107343777745525364, −8.529023279170600546505050969661, −7.994363092351022998772215333614, −6.91321893225085985679307279910, −5.34833338417124343163871268699, −4.95101317851646797674811810733, −3.24777707036220677111576390226, −0.46162728275807142523050137661,
2.12847128400288560045467415038, 3.15393371340108755715509399313, 4.66428313683191772468980219253, 5.82171309950913269172951484149, 7.24970103788360094043110081710, 8.513877431999114487586448624232, 9.619776391509290652470061825654, 10.42141685592739961671206799065, 11.39672935981650718414479216656, 11.95569431166868499636067262641