| L(s) = 1 | + (−2.59 − 0.551i)2-s + (0.494 − 1.65i)3-s + (4.59 + 2.04i)4-s + (−0.527 − 2.17i)5-s + (−2.19 + 4.03i)6-s + (1.02 − 1.77i)7-s + (−6.50 − 4.72i)8-s + (−2.51 − 1.64i)9-s + (0.171 + 5.92i)10-s + (5.61 + 1.19i)11-s + (5.67 − 6.61i)12-s + (2.04 − 0.434i)13-s + (−3.64 + 4.04i)14-s + (−3.86 − 0.199i)15-s + (7.52 + 8.35i)16-s + (−0.0247 − 0.0179i)17-s + ⋯ |
| L(s) = 1 | + (−1.83 − 0.389i)2-s + (0.285 − 0.958i)3-s + (2.29 + 1.02i)4-s + (−0.236 − 0.971i)5-s + (−0.897 + 1.64i)6-s + (0.387 − 0.671i)7-s + (−2.29 − 1.66i)8-s + (−0.836 − 0.547i)9-s + (0.0542 + 1.87i)10-s + (1.69 + 0.359i)11-s + (1.63 − 1.90i)12-s + (0.566 − 0.120i)13-s + (−0.973 + 1.08i)14-s + (−0.998 − 0.0514i)15-s + (1.88 + 2.08i)16-s + (−0.00600 − 0.00436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.178931 - 0.555540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178931 - 0.555540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.494 + 1.65i)T \) |
| 5 | \( 1 + (0.527 + 2.17i)T \) |
| good | 2 | \( 1 + (2.59 + 0.551i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.02 + 1.77i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.61 - 1.19i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 0.434i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (0.0247 + 0.0179i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.30 + 3.85i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.69 - 2.99i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.343 + 3.26i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.336 - 3.19i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.37 + 4.24i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.88 - 0.401i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.309 + 0.536i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.271 - 2.58i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (2.83 - 2.06i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-7.94 + 1.68i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-8.06 - 1.71i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.0735 + 0.699i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-3.81 + 2.76i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.55 - 7.85i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.25 + 11.9i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 5.35i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.99 - 6.15i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.06 - 10.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.69156257190769223788291114202, −10.94615247729355046831493551886, −9.524104150714064996696049346186, −8.857364195104520209246365679204, −8.148769270209427234103402815400, −7.22460025151014031687743927977, −6.33234112276851781504142019199, −3.90249489341025512363759826957, −1.89464024908760695146919868938, −0.884635212785618772684854660903,
2.12305066177227185258105402451, 3.77504291876164385821311569417, 5.97459036927068367218813567119, 6.69901604636732765605687379494, 8.199802625979683518413530887867, 8.654364473326359176897057719796, 9.621643721056607086889865215585, 10.45354231302725056035649431073, 11.25123744929891768821999868794, 11.85096317644983922942728050959
