| L(s) = 1 | + (−1.23 − 3.38i)2-s + (−3.55 − 0.626i)3-s + (−6.89 + 5.78i)4-s + (3.06 + 2.57i)5-s + (2.25 + 12.8i)6-s + (2.5 + 4.33i)7-s + (15.6 + 9.01i)8-s + (3.75 + 1.36i)9-s + (4.93 − 13.5i)10-s + (5 − 8.66i)11-s + (28.1 − 16.2i)12-s + (−3.55 + 0.626i)13-s + (11.5 − 13.8i)14-s + (−9.27 − 11.0i)15-s + (5.03 − 28.5i)16-s + (−14.0 + 5.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.616 − 1.69i)2-s + (−1.18 − 0.208i)3-s + (−1.72 + 1.44i)4-s + (0.612 + 0.514i)5-s + (0.376 + 2.13i)6-s + (0.357 + 0.618i)7-s + (1.95 + 1.12i)8-s + (0.417 + 0.152i)9-s + (0.493 − 1.35i)10-s + (0.454 − 0.787i)11-s + (2.34 − 1.35i)12-s + (−0.273 + 0.0481i)13-s + (0.827 − 0.986i)14-s + (−0.618 − 0.736i)15-s + (0.314 − 1.78i)16-s + (−0.829 + 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0362228 + 0.533600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0362228 + 0.533600i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (1.23 + 3.38i)T + (-3.06 + 2.57i)T^{2} \) |
| 3 | \( 1 + (3.55 + 0.626i)T + (8.45 + 3.07i)T^{2} \) |
| 5 | \( 1 + (-3.06 - 2.57i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.5 - 4.33i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-5 + 8.66i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.55 - 0.626i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (14.0 - 5.13i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (-26.8 + 22.4i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-6.16 + 16.9i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (31.2 - 18.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 21.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-35.5 - 6.26i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (15.3 + 12.8i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (9.39 + 3.42i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (48.6 + 58.0i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (6.16 + 16.9i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (30.6 - 25.7i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-13.5 + 37.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (69.5 - 82.8i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-18.2 + 103. i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (35.5 + 6.26i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-20 - 34.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (41.9 + 115. i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.03676913470524435877998579282, −10.20804043427720985727669152058, −9.116305213254168541256126059870, −8.458828499344158667474145257690, −6.77385791826083467738271954736, −5.77097573061392623648872438530, −4.55880985927964862929630543883, −3.01794385935880832763814329169, −1.88945947286758073381042486642, −0.41013427714309480359351376005,
1.17856042456487357358930992136, 4.53825975585422132934421885597, 5.12673198042168795216807528219, 5.99230939019544767727593993917, 6.93067899025164062377468161464, 7.62445623699582037208679524939, 9.010131956027353386515648428751, 9.514769558256469683252270943809, 10.57521824649279014252486280559, 11.45575247883987520724422047166
