L(s) = 1 | + 1.84·2-s + 1.39·4-s + (0.667 + 1.15i)5-s + (1.90 − 1.83i)7-s − 1.12·8-s + (1.22 + 2.12i)10-s + (0.756 − 1.31i)11-s + (−2.58 + 4.48i)13-s + (3.50 − 3.38i)14-s − 4.84·16-s + (−0.774 − 1.34i)17-s + (−1.25 + 2.16i)19-s + (0.927 + 1.60i)20-s + (1.39 − 2.41i)22-s + (−3.68 − 6.37i)23-s + ⋯ |
L(s) = 1 | + 1.30·2-s + 0.695·4-s + (0.298 + 0.516i)5-s + (0.719 − 0.694i)7-s − 0.396·8-s + (0.388 + 0.673i)10-s + (0.228 − 0.395i)11-s + (−0.717 + 1.24i)13-s + (0.936 − 0.904i)14-s − 1.21·16-s + (−0.187 − 0.325i)17-s + (−0.287 + 0.497i)19-s + (0.207 + 0.359i)20-s + (0.296 − 0.514i)22-s + (−0.767 − 1.32i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15929 + 0.0775675i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15929 + 0.0775675i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.90 + 1.83i)T \) |
good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 5 | \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.756 + 1.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.25 - 2.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 + (0.755 + 1.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + (2.80 + 4.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.68132403091468865552722704835, −11.76338862720031835608877717562, −10.94593661308677576117690380791, −9.775151985838219452618613454190, −8.467068523945794582047117828402, −7.01186987932375955997739123880, −6.18899723092342026367420100804, −4.78998226521223428600841376646, −3.99612609350324921865435446979, −2.38556131635350075837466356995,
2.28376037789533355383643127821, 3.88529083296901779370878286251, 5.25205267103630476225867562446, 5.57356098695714097027824682001, 7.21656457941564696111379988881, 8.558497574425318690071729101394, 9.512369323713665848353340920504, 10.91211513277231712574129966754, 12.05243608619042282350602575206, 12.57111981126569907674248360555