| L(s) = 1 | + (−2.43 + 1.08i)2-s + (−1.70 − 0.287i)3-s + (3.43 − 3.81i)4-s + (0.156 − 2.23i)5-s + (4.47 − 1.15i)6-s + (0.0677 − 0.117i)7-s + (−2.58 + 7.94i)8-s + (2.83 + 0.983i)9-s + (2.04 + 5.60i)10-s + (−3.84 + 1.71i)11-s + (−6.95 + 5.52i)12-s + (−1.38 − 0.617i)13-s + (−0.0378 + 0.359i)14-s + (−0.909 + 3.76i)15-s + (−1.25 − 11.9i)16-s + (1.49 − 4.58i)17-s + ⋯ |
| L(s) = 1 | + (−1.72 + 0.767i)2-s + (−0.986 − 0.166i)3-s + (1.71 − 1.90i)4-s + (0.0698 − 0.997i)5-s + (1.82 − 0.470i)6-s + (0.0256 − 0.0443i)7-s + (−0.912 + 2.80i)8-s + (0.944 + 0.327i)9-s + (0.645 + 1.77i)10-s + (−1.16 + 0.516i)11-s + (−2.00 + 1.59i)12-s + (−0.384 − 0.171i)13-s + (−0.0101 + 0.0961i)14-s + (−0.234 + 0.972i)15-s + (−0.314 − 2.99i)16-s + (0.361 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0102938 - 0.0538438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0102938 - 0.0538438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 + 0.287i)T \) |
| 5 | \( 1 + (-0.156 + 2.23i)T \) |
| good | 2 | \( 1 + (2.43 - 1.08i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (-0.0677 + 0.117i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.84 - 1.71i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.617i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.49 + 4.58i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.0886 - 0.272i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.581 - 5.53i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (4.07 + 0.866i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (9.47 - 2.01i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (8.14 - 5.91i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.23 + 2.77i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.81 - 1.44i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (1.22 + 3.77i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.28 - 1.01i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (0.993 - 0.442i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (8.73 - 1.85i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (0.994 + 3.06i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.15 + 1.56i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.63 + 0.773i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.682 + 0.758i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-5.75 - 4.17i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.06 + 0.439i)T + (88.6 + 39.4i)T^{2} \) |
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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
−11.56137757979138514902114945424, −10.49882767557592078258665113267, −9.797466475140417565130428805356, −8.916554864907291548054742945610, −7.60550758078758079756264890948, −7.24384437942451393664393579081, −5.64399043379744442083637352158, −5.14251617421948032979330931688, −1.72109553684791458601724822424, −0.084556594045430063486829413629,
2.10351340774122506560953203098, 3.58812679174571321620417348649, 5.77948330447672110555560525339, 6.98521548466508982072900191315, 7.76218269493238847751505756645, 8.988057679826599442311378451138, 10.22673846894615702021134759651, 10.55833704214559538431998669511, 11.20963305807833597064990800513, 12.20014724707186214355270533657
