| L(s) = 1 | + (−1.73 + i)2-s + (1.5 + 2.59i)3-s + (1.99 − 3.46i)4-s − 6.48i·5-s + (−5.19 − 3i)6-s + (−6.06 − 3.5i)7-s + 7.99i·8-s + (−4.5 + 7.79i)9-s + (6.48 + 11.2i)10-s + (21.2 − 12.2i)11-s + 12·12-s + (−45.5 + 10.8i)13-s + 14·14-s + (16.8 − 9.72i)15-s + (−8 − 13.8i)16-s + (−30.7 + 53.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.579i·5-s + (−0.353 − 0.204i)6-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.204 + 0.355i)10-s + (0.581 − 0.335i)11-s + 0.288·12-s + (−0.972 + 0.231i)13-s + 0.267·14-s + (0.289 − 0.167i)15-s + (−0.125 − 0.216i)16-s + (−0.438 + 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.411789689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411789689\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.73 - i)T \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (6.06 + 3.5i)T \) |
| 13 | \( 1 + (45.5 - 10.8i)T \) |
| good | 5 | \( 1 + 6.48iT - 125T^{2} \) |
| 11 | \( 1 + (-21.2 + 12.2i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (30.7 - 53.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-32.5 - 18.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (32.2 + 55.8i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-22.0 - 38.1i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 174. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-255. + 147. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-357. + 206. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 539. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 339.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-159. - 92.2i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-350. + 607. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-427. + 247. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (340. + 196. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 41.2iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 44.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.17e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (770. - 444. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-787. - 454. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.34172244675007348963223065248, −9.377067677522200110198513821858, −8.918389716645141374958608725879, −7.976586204361016091947074643371, −6.99227570751072569681355155123, −5.98059385479670095401279806674, −4.86940112536117283751977568034, −3.86736854333202394047282972371, −2.39191724309316543146623214079, −0.845712010236557051337096110179,
0.71701527729255921942924262460, 2.28194677945748502777054183095, 3.00822767232603620678696919158, 4.40837317280273879530448977430, 5.93259069402577144288683368060, 7.01269119191179470987721611004, 7.47538637903258565403815243217, 8.599718518975020298777881329897, 9.568316439587920363236424316402, 10.00095238197961406904922584927
