L(s) = 1 | + (−1.22 − 0.707i)2-s + (−0.866 − 1.5i)5-s + (1.62 + 2.09i)7-s + 2.82i·8-s + 2.44i·10-s + (−1.22 − 0.707i)11-s + (−2.12 + 1.22i)13-s + (−0.507 − 3.70i)14-s + (2.00 − 3.46i)16-s − 5.19·17-s + 7.34i·19-s + (0.999 + 1.73i)22-s + (−2.44 + 1.41i)23-s + (1 − 1.73i)25-s + 3.46·26-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (−0.387 − 0.670i)5-s + (0.612 + 0.790i)7-s + 0.999i·8-s + 0.774i·10-s + (−0.369 − 0.213i)11-s + (−0.588 + 0.339i)13-s + (−0.135 − 0.990i)14-s + (0.500 − 0.866i)16-s − 1.26·17-s + 1.68i·19-s + (0.213 + 0.369i)22-s + (−0.510 + 0.294i)23-s + (0.200 − 0.346i)25-s + 0.679·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.428035 + 0.236250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.428035 + 0.236250i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 2 | \( 1 + (1.22 + 0.707i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 7.34iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 - 1.41i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.12 - 3.53i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.12 - 1.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + (-4.33 - 7.5i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.33 - 7.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.12 + 1.22i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 1.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (14.8 + 8.57i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.84233615970941644462625347945, −9.917739517145756977964744314268, −9.132755063615372645901694892290, −8.307941249862875814870496173015, −7.928807865877351254151818670573, −6.28160658870295415905513030861, −5.19835378492439539028393475462, −4.41792260398365741808693741874, −2.58808962347771218959520633258, −1.45137577222642017933644219426,
0.38570348920692049076303913842, 2.54406677894802748445124348269, 3.98112849772319697581241016842, 4.87928147077297837399597071222, 6.59771358068851694556284817707, 7.17527069551474383855326753630, 7.85017781946836764007823643078, 8.714289240649755488326024573197, 9.632967775301316969994870824962, 10.59812505627237084478900161651