| L(s) = 1 | + 1.54i·5-s + 2.64i·7-s − 4.48·11-s + 1.54i·13-s − 23.6·17-s + 24.8·19-s − 35.2i·23-s + 22.5·25-s − 22.4i·29-s − 46.7i·31-s − 4.10·35-s + 58.5i·37-s + 26.9·41-s + 17.1·43-s + 36.1i·47-s + ⋯ |
| L(s) = 1 | + 0.309i·5-s + 0.377i·7-s − 0.407·11-s + 0.119i·13-s − 1.39·17-s + 1.30·19-s − 1.53i·23-s + 0.903·25-s − 0.774i·29-s − 1.50i·31-s − 0.117·35-s + 1.58i·37-s + 0.657·41-s + 0.399·43-s + 0.768i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.848524541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848524541\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 1.54iT - 25T^{2} \) |
| 11 | \( 1 + 4.48T + 121T^{2} \) |
| 13 | \( 1 - 1.54iT - 169T^{2} \) |
| 17 | \( 1 + 23.6T + 289T^{2} \) |
| 19 | \( 1 - 24.8T + 361T^{2} \) |
| 23 | \( 1 + 35.2iT - 529T^{2} \) |
| 29 | \( 1 + 22.4iT - 841T^{2} \) |
| 31 | \( 1 + 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 58.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 17.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 97.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 37.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 33.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 69.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 3.61T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 96.1T + 9.40e3T^{2} \) |
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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
−9.038745585280786078592757515675, −8.238227674385418669092818121821, −7.47679564536432607867933917393, −6.58138070069528642596737400786, −5.96921217836367218054909379179, −4.88833389682195114279930006533, −4.21695805374762711099339458163, −2.89554814856293800843005706157, −2.31013698089092948857975947442, −0.75524526519699635039338133128,
0.69154894167603822063880637661, 1.86761538093128545670378044296, 3.09288602277313798291250248819, 3.93546265980307474265407173445, 5.06637021885440238671822895099, 5.47523711090579121812834759080, 6.81233295357438995856908384951, 7.22859161976739210769165117238, 8.189521173731705043665092345393, 8.965972566976297667802747652928
