L(s) = 1 | + (−1.22 + 1.22i)3-s + (5.24 − 5.24i)5-s − 5.32·7-s − 2.99i·9-s + (−12.2 − 12.2i)11-s + (5.73 + 5.73i)13-s + 12.8i·15-s − 23.3·17-s + (−11.7 + 11.7i)19-s + (6.52 − 6.52i)21-s + 5.80·23-s − 29.9i·25-s + (3.67 + 3.67i)27-s + (−18.3 − 18.3i)29-s − 16.9i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (1.04 − 1.04i)5-s − 0.761·7-s − 0.333i·9-s + (−1.11 − 1.11i)11-s + (0.441 + 0.441i)13-s + 0.856i·15-s − 1.37·17-s + (−0.618 + 0.618i)19-s + (0.310 − 0.310i)21-s + 0.252·23-s − 1.19i·25-s + (0.136 + 0.136i)27-s + (−0.634 − 0.634i)29-s − 0.545i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.262002 - 0.666871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.262002 - 0.666871i\) |
\(L(2)\) |
|
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 | \( 1 + (-5.24 + 5.24i)T - 25iT^{2} \) |
| 7 | \( 1 + 5.32T + 49T^{2} \) |
| 11 | \( 1 + (12.2 + 12.2i)T + 121iT^{2} \) |
| 13 | \( 1 + (-5.73 - 5.73i)T + 169iT^{2} \) |
| 17 | \( 1 + 23.3T + 289T^{2} \) |
| 19 | \( 1 + (11.7 - 11.7i)T - 361iT^{2} \) |
| 23 | \( 1 - 5.80T + 529T^{2} \) |
| 29 | \( 1 + (18.3 + 18.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 16.9iT - 961T^{2} \) |
| 37 | \( 1 + (15.3 - 15.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 29.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (33.4 + 33.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-66.9 + 66.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-27.1 - 27.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (65.2 + 65.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-37.6 + 37.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 42.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 21.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (24.1 - 24.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 21.0T + 9.40e3T^{2} \) |
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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
−10.66310460021526518222472077902, −9.868860823817939208190448576508, −8.982707696234468293666268166521, −8.346049618066198008127204770316, −6.61923198416335759301920452262, −5.83084391758745833290415542522, −5.06045654769391180645973453835, −3.76673278538471484732923063799, −2.14155732216765573741427007777, −0.30346810089275018427742699678,
2.04685073136464876397095247133, 2.96122377467088860326528412472, 4.77428131761579149495989932216, 5.93981253533104984052366128284, 6.70270524608570399241387644547, 7.36691136205106887158097440978, 8.830539603663275659678213615329, 9.899849734375155634155365538694, 10.55744764492783132406891880406, 11.18727872831878551578453458204