L(s) = 1 | − 2.67·2-s − 2.89·3-s + 3.15·4-s + 3.34i·5-s + 7.74·6-s + (4.11 + 5.65i)7-s + 2.26·8-s − 0.605·9-s − 8.94i·10-s − 15.3i·11-s − 9.13·12-s + 7.80·13-s + (−11.0 − 15.1i)14-s − 9.68i·15-s − 18.6·16-s + 8.48·17-s + ⋯ |
L(s) = 1 | − 1.33·2-s − 0.965·3-s + 0.787·4-s + 0.668i·5-s + 1.29·6-s + (0.588 + 0.808i)7-s + 0.283·8-s − 0.0672·9-s − 0.894i·10-s − 1.39i·11-s − 0.761·12-s + 0.600·13-s + (−0.786 − 1.08i)14-s − 0.645i·15-s − 1.16·16-s + 0.499·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.263951 + 0.340457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.263951 + 0.340457i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-4.11 - 5.65i)T \) |
| 41 | \( 1 + (-38.1 - 15.1i)T \) |
good | 2 | \( 1 + 2.67T + 4T^{2} \) |
| 3 | \( 1 + 2.89T + 9T^{2} \) |
| 5 | \( 1 - 3.34iT - 25T^{2} \) |
| 11 | \( 1 + 15.3iT - 121T^{2} \) |
| 13 | \( 1 - 7.80T + 169T^{2} \) |
| 17 | \( 1 - 8.48T + 289T^{2} \) |
| 19 | \( 1 - 0.826T + 361T^{2} \) |
| 23 | \( 1 + 2.74T + 529T^{2} \) |
| 29 | \( 1 - 15.8iT - 841T^{2} \) |
| 31 | \( 1 - 52.1iT - 961T^{2} \) |
| 37 | \( 1 + 39.7T + 1.36e3T^{2} \) |
| 43 | \( 1 + 71.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 0.993T + 2.20e3T^{2} \) |
| 53 | \( 1 - 10.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.55iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 49.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 171.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 41.4T + 9.40e3T^{2} \) |
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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
−11.32778307855292670231284669416, −10.96685804401013697516519509469, −10.17539751112338287178462317841, −8.698843874925229469916725252677, −8.472769648164210626083451366783, −7.05163240866944319730128355663, −6.05267966958544152388805681495, −5.10391700354190514361048729801, −3.06939290789486192477964732520, −1.20480319439832146873323310580,
0.45471135450581538263085411534, 1.67413689427955200168937267822, 4.30447422478110096998866047116, 5.20331178737960668367634330872, 6.65155069245738258127321990012, 7.63938529362682197257411031433, 8.438569941261342772494572848516, 9.546201594260614472684506205986, 10.30780821531997052502157213915, 11.11268586485919010260502964904