| L(s) = 1 | + 1.41·2-s + (1.52 + 1.28i)3-s + 0.00332·4-s + (−0.713 − 1.23i)5-s + (2.16 + 1.81i)6-s + (0.0454 − 0.257i)7-s − 2.82·8-s + (0.168 + 0.958i)9-s + (−1.00 − 1.74i)10-s + (2.02 + 0.737i)11-s + (0.00508 + 0.00426i)12-s + (−4.11 + 1.49i)13-s + (0.0642 − 0.364i)14-s + (0.493 − 2.80i)15-s − 4.00·16-s + (0.200 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + (0.881 + 0.739i)3-s + 0.00166·4-s + (−0.318 − 0.552i)5-s + (0.882 + 0.740i)6-s + (0.0171 − 0.0973i)7-s − 0.999·8-s + (0.0563 + 0.319i)9-s + (−0.319 − 0.552i)10-s + (0.610 + 0.222i)11-s + (0.00146 + 0.00123i)12-s + (−1.14 + 0.415i)13-s + (0.0171 − 0.0974i)14-s + (0.127 − 0.722i)15-s − 1.00·16-s + (0.0486 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77458 + 0.301677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77458 + 0.301677i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 127 | \( 1 + (6.55 + 9.16i)T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + (-1.52 - 1.28i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (0.713 + 1.23i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.0454 + 0.257i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.02 - 0.737i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.11 - 1.49i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.200 - 1.13i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.19 + 3.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.98 - 1.45i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-6.72 - 2.44i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.08 - 1.75i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (6.78 - 5.68i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (3.56 - 2.99i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.60 - 1.34i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.716 - 1.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.943 - 0.791i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.444 - 2.52i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.08 + 1.87i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.94 + 11.0i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.06 - 11.7i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.60 + 9.71i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.617 + 0.517i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.38 + 0.502i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.57 - 9.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.73 + 15.4i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.77543211884170050404945991244, −12.37057502236804170961738926124, −11.88834435171949029433172313762, −10.05245851005910487973648807338, −9.194926207654232226276586565481, −8.351000553836111515185017632404, −6.67223742804114727974773409239, −4.92887691017449730616680928676, −4.24111066833671425629751168596, −2.99221149919902102138180489888,
2.58374305500955389696716987134, 3.73745271720673371093107426908, 5.30023221362157296043288061733, 6.73692347854759512273052457322, 7.80790768480197434519527164788, 8.888905971472865826803837864347, 10.18567135243159167739211002556, 11.84176983596676900208738312959, 12.40500905674247798746866985513, 13.53975064372687113288272297457
