| L(s) = 1 | + (1.31 − 2.69i)3-s + (−1.53 + 4.75i)5-s + 12.2i·7-s + (−5.56 − 7.07i)9-s + 3.02·11-s − 13.7·13-s + (10.8 + 10.3i)15-s + 25.8·17-s − 14.0i·19-s + (33.0 + 16.0i)21-s − 17.8·23-s + (−20.2 − 14.6i)25-s + (−26.3 + 5.72i)27-s − 27.5·29-s − 33.8·31-s + ⋯ |
| L(s) = 1 | + (0.437 − 0.899i)3-s + (−0.307 + 0.951i)5-s + 1.74i·7-s + (−0.617 − 0.786i)9-s + 0.274·11-s − 1.05·13-s + (0.721 + 0.692i)15-s + 1.52·17-s − 0.741i·19-s + (1.57 + 0.764i)21-s − 0.775·23-s + (−0.810 − 0.585i)25-s + (−0.977 + 0.211i)27-s − 0.951·29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6261464153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6261464153\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.31 + 2.69i)T \) |
| 5 | \( 1 + (1.53 - 4.75i)T \) |
| good | 7 | \( 1 - 12.2iT - 49T^{2} \) |
| 11 | \( 1 - 3.02T + 121T^{2} \) |
| 13 | \( 1 + 13.7T + 169T^{2} \) |
| 17 | \( 1 - 25.8T + 289T^{2} \) |
| 19 | \( 1 + 14.0iT - 361T^{2} \) |
| 23 | \( 1 + 17.8T + 529T^{2} \) |
| 29 | \( 1 + 27.5T + 841T^{2} \) |
| 31 | \( 1 + 33.8T + 961T^{2} \) |
| 37 | \( 1 - 14.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 63.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 28.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 2.88iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 69.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 69.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 109.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 16.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 66.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62.9iT - 9.40e3T^{2} \) |
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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
−9.889410239441002859570512620023, −9.385037130615246593007389286438, −8.316113822789428461236809808951, −7.72123751486495043454669626970, −6.84312502350793042646713026927, −6.01870692200672458700399344533, −5.21280771281305390958888628243, −3.47611789178857755730402292986, −2.71383821873310026666090806911, −1.87119038138836237301904140634,
0.17702534501530998113778438356, 1.64333057179192287632705557864, 3.49908840732339903466954029917, 3.98239463094575776532032545021, 4.88020815083440337702687167242, 5.72723564265631979814116835229, 7.43338813609256387246234939439, 7.67278717741416552841148721765, 8.692279884280315884704706421990, 9.769261779946648397543071317410
