L(s) = 1 | + (−1.69 + 0.348i)3-s + (−2.08 + 1.20i)5-s + (2.53 + 0.763i)7-s + (2.75 − 1.18i)9-s + (−2.81 − 1.62i)11-s + (4.35 + 2.51i)13-s + (3.11 − 2.76i)15-s + (0.795 − 0.459i)17-s + (3.22 − 5.57i)19-s + (−4.56 − 0.414i)21-s + (−5.31 + 3.06i)23-s + (0.399 − 0.691i)25-s + (−4.26 + 2.96i)27-s + (1.22 + 2.12i)29-s + 3.21·31-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.200i)3-s + (−0.932 + 0.538i)5-s + (0.957 + 0.288i)7-s + (0.919 − 0.393i)9-s + (−0.847 − 0.489i)11-s + (1.20 + 0.698i)13-s + (0.805 − 0.714i)15-s + (0.192 − 0.111i)17-s + (0.739 − 1.28i)19-s + (−0.995 − 0.0904i)21-s + (−1.10 + 0.640i)23-s + (0.0798 − 0.138i)25-s + (−0.821 + 0.570i)27-s + (0.228 + 0.394i)29-s + 0.578·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8664682375\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8664682375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 - 0.348i)T \) |
| 7 | \( 1 + (-2.53 - 0.763i)T \) |
good | 5 | \( 1 + (2.08 - 1.20i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.81 + 1.62i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.35 - 2.51i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.795 + 0.459i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.22 + 5.57i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.31 - 3.06i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 + (4.08 - 7.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.3 - 5.99i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (10.2 - 5.89i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 + (1.56 + 2.71i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 - 9.49iT - 61T^{2} \) |
| 67 | \( 1 + 0.359iT - 67T^{2} \) |
| 71 | \( 1 - 2.32iT - 71T^{2} \) |
| 73 | \( 1 + (4.01 - 2.31i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 4.36iT - 79T^{2} \) |
| 83 | \( 1 + (-8.68 - 15.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.68 + 5.59i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.79 - 5.65i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.48835491323687837916382615117, −9.449119141189484413585116176079, −8.357892142538074229135007159468, −7.69361172665509094181962873564, −6.78946622087106912374596874862, −5.86432827032481994168888488925, −4.97250855899722082445955205864, −4.13882860723992589262594469032, −3.02951726502967893571978771859, −1.26464407738887920873965617016,
0.52391948707560138279055760886, 1.80166552788517007403759235166, 3.74670448800055084232511363859, 4.49497765184846502030735113863, 5.41340722487372790637895303633, 6.11977647628243919656821080892, 7.58471450421543538075519502004, 7.80833185382898332774378445418, 8.623137313274789366214682334988, 10.13816276135954160296588761322