L(s) = 1 | − 2-s + 4-s + (0.230 + 0.398i)5-s − 8-s + (−0.230 − 0.398i)10-s + (−1.82 + 3.15i)11-s + (−0.730 + 1.26i)13-s + 16-s + (−1.86 − 3.23i)17-s + (2.02 − 3.51i)19-s + (0.230 + 0.398i)20-s + (1.82 − 3.15i)22-s + (0.566 + 0.981i)23-s + (2.39 − 4.14i)25-s + (0.730 − 1.26i)26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.102 + 0.178i)5-s − 0.353·8-s + (−0.0728 − 0.126i)10-s + (−0.549 + 0.952i)11-s + (−0.202 + 0.350i)13-s + 0.250·16-s + (−0.452 − 0.784i)17-s + (0.465 − 0.805i)19-s + (0.0514 + 0.0891i)20-s + (0.388 − 0.673i)22-s + (0.118 + 0.204i)23-s + (0.478 − 0.829i)25-s + (0.143 − 0.248i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6842878831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6842878831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.230 - 0.398i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.82 - 3.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.730 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.566 - 0.981i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.514T + 31T^{2} \) |
| 37 | \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.472 - 0.819i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + (6.21 + 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + (-6.62 - 11.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.59 - 9.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.233918044886156131568673198313, −8.406830519819940471203113918048, −7.59521621936380890567008523881, −6.90709617368077337434257984911, −6.41224418327312286793777488051, −5.03962237556822855275004232334, −4.65817091737084468178588731235, −3.12210553020404409762325861954, −2.45913456788336638515630413006, −1.26882124729969113457086842577,
0.29422409385035194604713376337, 1.56558449266887427138845847381, 2.69847925047286153055131521893, 3.57519086952350046865901301285, 4.69671052840012894567017717694, 5.80724973152920837906777477878, 6.12823734447895059823969343674, 7.38779601478677410705334923970, 7.83590870732858212857889205882, 8.706911801520283046532522575713