L(s) = 1 | + (1.22 + 1.22i)3-s + (2 + 2i)5-s − 0.378·7-s + 2.99i·9-s + (−10.5 + 10.5i)11-s + (−6.46 + 6.46i)13-s + 4.89i·15-s − 11.8·17-s + (−18.6 − 18.6i)19-s + (−0.464 − 0.464i)21-s − 12.0·23-s − 17i·25-s + (−3.67 + 3.67i)27-s + (−1.07 + 1.07i)29-s + 6.38i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (0.400 + 0.400i)5-s − 0.0541·7-s + 0.333i·9-s + (−0.959 + 0.959i)11-s + (−0.497 + 0.497i)13-s + 0.326i·15-s − 0.697·17-s + (−0.982 − 0.982i)19-s + (−0.0221 − 0.0221i)21-s − 0.524·23-s − 0.680i·25-s + (−0.136 + 0.136i)27-s + (−0.0369 + 0.0369i)29-s + 0.206i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8485074062\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8485074062\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
good | 5 | \( 1 + (-2 - 2i)T + 25iT^{2} \) |
| 7 | \( 1 + 0.378T + 49T^{2} \) |
| 11 | \( 1 + (10.5 - 10.5i)T - 121iT^{2} \) |
| 13 | \( 1 + (6.46 - 6.46i)T - 169iT^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 19 | \( 1 + (18.6 + 18.6i)T + 361iT^{2} \) |
| 23 | \( 1 + 12.0T + 529T^{2} \) |
| 29 | \( 1 + (1.07 - 1.07i)T - 841iT^{2} \) |
| 31 | \( 1 - 6.38iT - 961T^{2} \) |
| 37 | \( 1 + (-35.3 - 35.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 3.56iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (29.9 - 29.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 31.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-68.7 - 68.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (27.8 - 27.8i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-62.8 + 62.8i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (30.5 + 30.5i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 131.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 50.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 114. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (1.92 + 1.92i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 74iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 89.2T + 9.40e3T^{2} \) |
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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
−10.32523119365672162685702336701, −9.815514833816990414374304361505, −8.910976118708834506575413141904, −8.012506378130958775647544872941, −7.04478087984595838007960830702, −6.26248202084933565769396821877, −4.90284103274652179764682731393, −4.32157172625163660112676052283, −2.76810343613135891612593850747, −2.10211232045356271446609937970,
0.24713648153737983750409689134, 1.85914392213555697752655597435, 2.89680500030943326923923493041, 4.11533128096452411704130425649, 5.41281303401873010378218706462, 6.05243071047194644431183027841, 7.22498586732214084217697822261, 8.147215303282099546827903545353, 8.668248094683037562507987373692, 9.707716905562012803048121090142