| L(s) = 1 | − 6.56·5-s − 11.5i·7-s + 5.05i·13-s − 8.56i·17-s − 10.8i·19-s − 11.7·23-s + 18.0·25-s + 20.2i·29-s − 36.4·31-s + 75.5i·35-s − 36.8·37-s + 42.3i·41-s − 72.4i·43-s + 10.9·47-s − 83.6·49-s + ⋯ |
| L(s) = 1 | − 1.31·5-s − 1.64i·7-s + 0.389i·13-s − 0.504i·17-s − 0.570i·19-s − 0.510·23-s + 0.721·25-s + 0.697i·29-s − 1.17·31-s + 2.15i·35-s − 0.997·37-s + 1.03i·41-s − 1.68i·43-s + 0.232·47-s − 1.70·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2743433298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2743433298\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 6.56T + 25T^{2} \) |
| 7 | \( 1 + 11.5iT - 49T^{2} \) |
| 13 | \( 1 - 5.05iT - 169T^{2} \) |
| 17 | \( 1 + 8.56iT - 289T^{2} \) |
| 19 | \( 1 + 10.8iT - 361T^{2} \) |
| 23 | \( 1 + 11.7T + 529T^{2} \) |
| 29 | \( 1 - 20.2iT - 841T^{2} \) |
| 31 | \( 1 + 36.4T + 961T^{2} \) |
| 37 | \( 1 + 36.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 72.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 72.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47T + 3.48e3T^{2} \) |
| 61 | \( 1 + 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 37.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 24.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 71.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 11.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 75.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2.98T + 9.40e3T^{2} \) |
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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
−8.167918440104545316055331301966, −7.45405689724060638573068252589, −7.15202150483318198839709867420, −6.42006311142863813175176272504, −5.10580479551838871649514748182, −4.49249984088769181422570579341, −3.73111632711267677045597683647, −3.30081068529877023538178196721, −1.80110107862901695860885679033, −0.63041656370904716051226782672,
0.088024883767239922842884979323, 1.64497299179333872950345306588, 2.62379495013134402207503821783, 3.49128411267255339171521633896, 4.17877626655380186430363797038, 5.18355990058950272631526837400, 5.81028599459486219932118310791, 6.52742468030157312540256574905, 7.65293222601953339497315674016, 7.978299954188661145797516761261
