| L(s) = 1 | + (−4.03 − 2.95i)5-s − 7.07i·7-s + 9.43i·11-s + 6.86i·13-s + 6.05·17-s + 14.3·19-s − 11.4·23-s + (7.55 + 23.8i)25-s − 21.2i·29-s + 8.47·31-s + (−20.8 + 28.5i)35-s − 22.0i·37-s + 58.8i·41-s + 49.4i·43-s − 49.9·47-s + ⋯ |
| L(s) = 1 | + (−0.806 − 0.590i)5-s − 1.01i·7-s + 0.857i·11-s + 0.528i·13-s + 0.356·17-s + 0.754·19-s − 0.499·23-s + (0.302 + 0.953i)25-s − 0.731i·29-s + 0.273·31-s + (−0.596 + 0.815i)35-s − 0.595i·37-s + 1.43i·41-s + 1.14i·43-s − 1.06·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.576109934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576109934\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.03 + 2.95i)T \) |
| good | 7 | \( 1 + 7.07iT - 49T^{2} \) |
| 11 | \( 1 - 9.43iT - 121T^{2} \) |
| 13 | \( 1 - 6.86iT - 169T^{2} \) |
| 17 | \( 1 - 6.05T + 289T^{2} \) |
| 19 | \( 1 - 14.3T + 361T^{2} \) |
| 23 | \( 1 + 11.4T + 529T^{2} \) |
| 29 | \( 1 + 21.2iT - 841T^{2} \) |
| 31 | \( 1 - 8.47T + 961T^{2} \) |
| 37 | \( 1 + 22.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 69.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 50.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 92.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 42.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 84.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 108. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 23.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 73.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 69.4iT - 9.40e3T^{2} \) |
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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
−8.739976949290723049232254117541, −7.88808686944477107569743174457, −7.40963615252807066593592048371, −6.67274896704666854417022919805, −5.54083473880432191463416211298, −4.49693534520114803543079063565, −4.15932536083997618103564692248, −3.09926694274561557231611334517, −1.65378008342917585362840011614, −0.62611284965658869552345471706,
0.71118085802800215924709131197, 2.29908172661763065388746246672, 3.19957484482877561748838747235, 3.82947848394493303875506986097, 5.18028230131197651955438396623, 5.71628292485103925466863016434, 6.68576327157238005030849313294, 7.45050410339879816326074856229, 8.341248841939898452455308136920, 8.694884605917312469898442154010
