| L(s) = 1 | + (1.15 − 1.27i)2-s + (1.47 + 0.903i)3-s + (−0.100 − 0.957i)4-s + (−0.0644 − 2.23i)5-s + (2.85 − 0.850i)6-s + (−2.11 − 3.66i)7-s + (1.44 + 1.04i)8-s + (1.36 + 2.66i)9-s + (−2.93 − 2.49i)10-s + (−3.66 + 4.06i)11-s + (0.716 − 1.50i)12-s + (1.55 + 1.72i)13-s + (−7.12 − 1.51i)14-s + (1.92 − 3.36i)15-s + (4.88 − 1.03i)16-s + (−1.13 − 0.824i)17-s + ⋯ |
| L(s) = 1 | + (0.814 − 0.904i)2-s + (0.853 + 0.521i)3-s + (−0.0503 − 0.478i)4-s + (−0.0288 − 0.999i)5-s + (1.16 − 0.347i)6-s + (−0.799 − 1.38i)7-s + (0.510 + 0.370i)8-s + (0.456 + 0.889i)9-s + (−0.927 − 0.788i)10-s + (−1.10 + 1.22i)11-s + (0.206 − 0.434i)12-s + (0.431 + 0.478i)13-s + (−1.90 − 0.404i)14-s + (0.496 − 0.867i)15-s + (1.22 − 0.259i)16-s + (−0.275 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90211 - 1.05361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90211 - 1.05361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.47 - 0.903i)T \) |
| 5 | \( 1 + (0.0644 + 2.23i)T \) |
| good | 2 | \( 1 + (-1.15 + 1.27i)T + (-0.209 - 1.98i)T^{2} \) |
| 7 | \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.66 - 4.06i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.55 - 1.72i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.13 + 0.824i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.74 - 3.45i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.41 + 0.726i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (3.94 + 1.75i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (2.89 - 1.28i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.930 + 2.86i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.09 - 3.43i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.62 + 1.16i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.61 + 3.35i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (7.75 + 8.61i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-7.25 + 8.05i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 4.58i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (5.34 - 3.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.20 - 3.69i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.62 + 1.16i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.713 - 6.78i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (1.41 + 4.35i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-12.6 - 5.62i)T + (64.9 + 72.0i)T^{2} \) |
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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
−12.44361240246987878964965425766, −11.13195197147783510507880114970, −10.06790988499255608985643483037, −9.629143871262150432493659084826, −8.073726159713496517489420787773, −7.34094840589098249359277247128, −5.19657709394404857254276113224, −4.24245602483075843639557554690, −3.54505425305074645745378240774, −1.94276370219940752199903708705,
2.68323450778984058511165802716, 3.49574025136518732176748861282, 5.60225390871809388130699446365, 6.13197694840488787878584809280, 7.23704928900070193925320220404, 8.134807015494960988261448383009, 9.261376254952390761364148764457, 10.41994804142155165458001378603, 11.67288268187608480851797668143, 12.98704819383540307725663139673
