L(s) = 1 | + (−0.527 − 1.62i)2-s + (−1.85 − 1.34i)3-s + (−0.739 + 0.537i)4-s + (1.25 − 3.86i)5-s + (−1.20 + 3.72i)6-s + (−0.809 + 0.587i)7-s + (−1.49 − 1.08i)8-s + (0.695 + 2.14i)9-s − 6.94·10-s + 2.09·12-s + (1.01 + 3.11i)13-s + (1.38 + 1.00i)14-s + (−7.53 + 5.47i)15-s + (−1.54 + 4.74i)16-s + (−0.408 + 1.25i)17-s + (3.10 − 2.25i)18-s + ⋯ |
L(s) = 1 | + (−0.373 − 1.14i)2-s + (−1.07 − 0.777i)3-s + (−0.369 + 0.268i)4-s + (0.561 − 1.72i)5-s + (−0.493 + 1.51i)6-s + (−0.305 + 0.222i)7-s + (−0.530 − 0.385i)8-s + (0.231 + 0.713i)9-s − 2.19·10-s + 0.604·12-s + (0.280 + 0.864i)13-s + (0.369 + 0.268i)14-s + (−1.94 + 1.41i)15-s + (−0.385 + 1.18i)16-s + (−0.0989 + 0.304i)17-s + (0.732 − 0.532i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.381502 + 0.199697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.381502 + 0.199697i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.527 + 1.62i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.85 + 1.34i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.25 + 3.86i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 3.11i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.408 - 1.25i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.74 + 1.27i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + (-0.197 + 0.143i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.12 + 6.53i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.206 - 0.150i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.63 - 3.36i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + (-3.29 - 2.39i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.54 + 4.76i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.795 + 0.578i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.570 + 1.75i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 3.00T + 67T^{2} \) |
| 71 | \( 1 + (-2.00 + 6.15i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.80 + 5.67i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.71 + 5.26i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.639 + 1.96i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + (0.798 + 2.45i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.400344515353002135664449621416, −9.075174622355936138347163549401, −8.010778331686728892205664202575, −6.51213482764678882424187599857, −6.04370453390874349664861543988, −5.05005068353400131380094588469, −3.95843256689263784252028121459, −2.13313263266252753671109720313, −1.36396288889547340383453589967, −0.27555341394913068575234289006,
2.61539950369027199289609286810, 3.69530826782603343624911108866, 5.24329939108626263257615692626, 5.89405728710968228719891133883, 6.54836278228664380570109145777, 7.17973179473941673994219241330, 8.138928211129488153474818066048, 9.367495797057436713211965949244, 10.22795628850665397379990976906, 10.64871608353858614203010600787