L(s) = 1 | − 1.73i·3-s + (0.5 − 0.133i)5-s + (2.13 − 1.23i)7-s − 2.99·9-s + (−0.133 + 0.5i)11-s + (−1.23 − 4.59i)13-s + (−0.232 − 0.866i)15-s + 4·17-s + (3 − 3i)19-s + (−2.13 − 3.69i)21-s + (−0.401 − 0.232i)23-s + (−4.09 + 2.36i)25-s + 5.19i·27-s + (−3.23 − 0.866i)29-s + (−0.598 + 1.03i)31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (0.223 − 0.0599i)5-s + (0.806 − 0.465i)7-s − 0.999·9-s + (−0.0403 + 0.150i)11-s + (−0.341 − 1.27i)13-s + (−0.0599 − 0.223i)15-s + 0.970·17-s + (0.688 − 0.688i)19-s + (−0.465 − 0.806i)21-s + (−0.0838 − 0.0483i)23-s + (−0.819 + 0.473i)25-s + 0.999i·27-s + (−0.600 − 0.160i)29-s + (−0.107 + 0.186i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.940047 - 1.17127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.940047 - 1.17127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
good | 5 | \( 1 + (-0.5 + 0.133i)T + (4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.13 + 1.23i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.133 - 0.5i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.23 + 4.59i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (-3 + 3i)T - 19iT^{2} \) |
| 23 | \( 1 + (0.401 + 0.232i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.23 + 0.866i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (0.598 - 1.03i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.73 + 7.73i)T + 37iT^{2} \) |
| 41 | \( 1 + (-9.69 - 5.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.33 + 8.69i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.59 - 1.5i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-14.4 - 3.86i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-0.330 - 1.23i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + 0.535iT - 73T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.7 - 3.16i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 11.8iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.59911754776883201813314409801, −9.599281429229323555327424371645, −8.483462672364116091131844436598, −7.60151306850621739276078406874, −7.22627305520648180321460997345, −5.74145650076230080048615771115, −5.20466998171835004367448205588, −3.56018545535216238145527784744, −2.24284944436441471978316693036, −0.896052327697105328122559191309,
1.91409019923414559038292049707, 3.34440003674585134994328785119, 4.45559449432927182013915747684, 5.33866764512297021622194394154, 6.17218802157949538224783852568, 7.61485953140342362977510081610, 8.433926509439944936160982320656, 9.438222684118417926292826791476, 9.910144076696271561552479803524, 11.01646988812635477903815080031