L(s) = 1 | + 1.27i·2-s + (−0.583 + 1.01i)3-s + 0.370·4-s + (−1.57 − 0.907i)5-s + (−1.29 − 0.745i)6-s + 3.02i·8-s + (0.817 + 1.41i)9-s + (1.15 − 2.00i)10-s + (2.40 + 1.38i)11-s + (−0.216 + 0.374i)12-s + (3.58 + 0.402i)13-s + (1.83 − 1.05i)15-s − 3.12·16-s + 2.74·17-s + (−1.80 + 1.04i)18-s + (−5.08 + 2.93i)19-s + ⋯ |
L(s) = 1 | + 0.902i·2-s + (−0.337 + 0.583i)3-s + 0.185·4-s + (−0.702 − 0.405i)5-s + (−0.527 − 0.304i)6-s + 1.06i·8-s + (0.272 + 0.472i)9-s + (0.366 − 0.634i)10-s + (0.725 + 0.418i)11-s + (−0.0624 + 0.108i)12-s + (0.993 + 0.111i)13-s + (0.473 − 0.273i)15-s − 0.780·16-s + 0.665·17-s + (−0.426 + 0.246i)18-s + (−1.16 + 0.673i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.252077 + 1.26071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.252077 + 1.26071i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.58 - 0.402i)T \) |
good | 2 | \( 1 - 1.27iT - 2T^{2} \) |
| 3 | \( 1 + (0.583 - 1.01i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.57 + 0.907i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 + (5.08 - 2.93i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.79 - 1.03i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.74iT - 37T^{2} \) |
| 41 | \( 1 + (5.51 - 3.18i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.76 - 3.32i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 9.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 3.07iT - 59T^{2} \) |
| 61 | \( 1 + (-0.540 - 0.936i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.34 + 2.50i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.35 - 1.35i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.64 + 3.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.86 + 13.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.97iT - 83T^{2} \) |
| 89 | \( 1 - 16.0iT - 89T^{2} \) |
| 97 | \( 1 + (-12.3 - 7.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.80559714721814563923901429153, −10.27205735268847772830881384449, −8.963858678057750085071769265745, −8.174654111036473461785807778020, −7.50646935550833828931689104014, −6.39596787488319186585940143294, −5.68410459753261740800968538790, −4.52590097769786824750339231136, −3.81329651604730884510982313762, −1.85373254220572528227519880723,
0.74654564890158799532617812296, 2.04473576669002256554009186420, 3.55191741944683488981942877706, 4.00328618217459018221643632827, 5.98413851103102059266273010787, 6.57351112823867322724584107200, 7.43665299025909111235885632273, 8.462333976981500583302919441003, 9.571165301691853602903909469256, 10.46825590203539907769822152079