L(s) = 1 | + 3.80·3-s − 2.23i·5-s − 8.50i·7-s + 5.47·9-s − 1.79·11-s + 0.472i·13-s − 8.50i·15-s − 23.8·17-s − 9.40·19-s − 32.3i·21-s − 16.1i·23-s − 5.00·25-s − 13.4·27-s + 6.94i·29-s − 47.4i·31-s + ⋯ |
L(s) = 1 | + 1.26·3-s − 0.447i·5-s − 1.21i·7-s + 0.608·9-s − 0.163·11-s + 0.0363i·13-s − 0.567i·15-s − 1.40·17-s − 0.494·19-s − 1.54i·21-s − 0.700i·23-s − 0.200·25-s − 0.497·27-s + 0.239i·29-s − 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.909150740\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.909150740\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
good | 3 | \( 1 - 3.80T + 9T^{2} \) |
| 7 | \( 1 + 8.50iT - 49T^{2} \) |
| 11 | \( 1 + 1.79T + 121T^{2} \) |
| 13 | \( 1 - 0.472iT - 169T^{2} \) |
| 17 | \( 1 + 23.8T + 289T^{2} \) |
| 19 | \( 1 + 9.40T + 361T^{2} \) |
| 23 | \( 1 + 16.1iT - 529T^{2} \) |
| 29 | \( 1 - 6.94iT - 841T^{2} \) |
| 31 | \( 1 + 47.4iT - 961T^{2} \) |
| 37 | \( 1 + 26.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.00T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 88.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1T + 9.40e3T^{2} \) |
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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
−9.113398322480063835602541521494, −8.379807139878851373595561819414, −7.69953226916822218456162875064, −6.95269812316936743056522952803, −5.90943233583833475794068590968, −4.40480350538477399313455528400, −4.11286696181439445094872400203, −2.84914290736239458010771211549, −1.90795510445816607268416886390, −0.41809923654717290356479194978,
1.90456168102444421342172914644, 2.61348083326064778858670369020, 3.39445576756201181337833047724, 4.54642062689183321948655029992, 5.66008632729104372694053686946, 6.56897317105693200936076003356, 7.47794515295246965407551217506, 8.432288074081293304093289819627, 8.836678070942861593656918582436, 9.502152503252914622471522143628