L(s) = 1 | − 2-s + (0.5 + 1.65i)3-s + 4-s + (−2.18 − 3.78i)5-s + (−0.5 − 1.65i)6-s − 8-s + (−2.5 + 1.65i)9-s + (2.18 + 3.78i)10-s + (0.686 − 1.18i)11-s + (0.5 + 1.65i)12-s + (−1 + 1.73i)13-s + (5.18 − 5.51i)15-s + 16-s + (−0.686 − 1.18i)17-s + (2.5 − 1.65i)18-s + (−2.5 + 4.33i)19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 + 0.957i)3-s + 0.5·4-s + (−0.977 − 1.69i)5-s + (−0.204 − 0.677i)6-s − 0.353·8-s + (−0.833 + 0.552i)9-s + (0.691 + 1.19i)10-s + (0.206 − 0.358i)11-s + (0.144 + 0.478i)12-s + (−0.277 + 0.480i)13-s + (1.33 − 1.42i)15-s + 0.250·16-s + (−0.166 − 0.288i)17-s + (0.589 − 0.390i)18-s + (−0.573 + 0.993i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.212352 + 0.419431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.212352 + 0.419431i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 - 1.65i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2.18 + 3.78i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.686 + 1.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.686 + 1.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.813 - 1.40i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.37 - 7.57i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.31 - 4.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.05 - 7.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-4.37 - 7.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.11T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + (6.05 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 + (8.74 + 15.1i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.05 - 7.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.25654364590190934890442428196, −9.323766023920277870438810432577, −8.810065733564721948286372665220, −8.255647051388552304519508352129, −7.45212130261850003651368919876, −5.97922389491566618933190716438, −4.88131699890072881416083461128, −4.26668683659216218235463115051, −3.17659171318759441331865818555, −1.38059435506006490010756870256,
0.28275972629578904859996732317, 2.28577084803254638252556945450, 2.93794216901806269454716678468, 4.11883595147716406648718460210, 5.99163455981608958712422138955, 6.85177736274537856809789061654, 7.20522880343029496510321895500, 8.050000133186748098640191465932, 8.718501963835645491466014503024, 9.988522548724797040907127724231