L(s) = 1 | + (1.19 + 2.06i)2-s + (−1.37 − 2.38i)3-s + (−1.85 + 3.20i)4-s − 0.982·5-s + (3.28 − 5.69i)6-s − 4.06·8-s + (−2.28 + 3.95i)9-s + (−1.17 − 2.03i)10-s + (0.293 + 0.509i)11-s + 10.1·12-s + (−2.39 + 2.69i)13-s + (1.35 + 2.34i)15-s + (−1.15 − 1.99i)16-s + (−3.22 + 5.58i)17-s − 10.9·18-s + (−1.91 + 3.31i)19-s + ⋯ |
L(s) = 1 | + (0.844 + 1.46i)2-s + (−0.794 − 1.37i)3-s + (−0.925 + 1.60i)4-s − 0.439·5-s + (1.34 − 2.32i)6-s − 1.43·8-s + (−0.761 + 1.31i)9-s + (−0.370 − 0.642i)10-s + (0.0886 + 0.153i)11-s + 2.94·12-s + (−0.663 + 0.748i)13-s + (0.348 + 0.604i)15-s + (−0.288 − 0.498i)16-s + (−0.782 + 1.35i)17-s − 2.57·18-s + (−0.438 + 0.760i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0237770 - 0.716696i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0237770 - 0.716696i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.39 - 2.69i)T \) |
good | 2 | \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.37 + 2.38i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.982T + 5T^{2} \) |
| 11 | \( 1 + (-0.293 - 0.509i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 - 3.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.13 + 7.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.98 - 3.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + (0.877 + 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.83 - 3.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.19 - 5.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 0.425T + 53T^{2} \) |
| 59 | \( 1 + (-3.00 + 5.20i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.10 + 1.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.80 - 3.11i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 + (1.04 + 1.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.84 + 6.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.41802199117222291063904971497, −10.28768314508220989530270103726, −8.612414651379199301256497090486, −7.978191722451259856812790759156, −7.22900327588305698938840646427, −6.29574932248340982513244259816, −6.18561908721863658633074669499, −4.80668824796761430156960131097, −4.00601702728742262940727639712, −1.97773268463008196544616204740,
0.31277695431341298370700645824, 2.50399705313134471980680001288, 3.60963236269781726953614217277, 4.39109624051100852605796028385, 5.08094315472615074688098133796, 5.86489852662051966639061862651, 7.43361678762777666717330864909, 8.979458381950145459110378478650, 9.845584340106662003681702256961, 10.27255304313379267991831627645