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
−11.45575247883987520724422047166, −10.57521824649279014252486280559, −9.514769558256469683252270943809, −9.010131956027353386515648428751, −7.62445623699582037208679524939, −6.93067899025164062377468161464, −5.99230939019544767727593993917, −5.12673198042168795216807528219, −4.53825975585422132934421885597, −1.17856042456487357358930992136,
0.41013427714309480359351376005, 1.88945947286758073381042486642, 3.01794385935880832763814329169, 4.55880985927964862929630543883, 5.77097573061392623648872438530, 6.77385791826083467738271954736, 8.458828499344158667474145257690, 9.116305213254168541256126059870, 10.20804043427720985727669152058, 11.03676913470524435877998579282