L(s) = 1 | + (−0.909 + 0.909i)5-s + 0.654·7-s + (−13.3 − 13.3i)11-s + (8.32 + 8.32i)13-s + 3.93·17-s + (−16.8 + 16.8i)19-s − 23.1·23-s + 23.3i·25-s + (−35.6 − 35.6i)29-s − 45.5i·31-s + (−0.595 + 0.595i)35-s + (10.1 − 10.1i)37-s + 28.4i·41-s + (−22.7 − 22.7i)43-s + 10.7i·47-s + ⋯ |
L(s) = 1 | + (−0.181 + 0.181i)5-s + 0.0935·7-s + (−1.21 − 1.21i)11-s + (0.640 + 0.640i)13-s + 0.231·17-s + (−0.889 + 0.889i)19-s − 1.00·23-s + 0.933i·25-s + (−1.22 − 1.22i)29-s − 1.46i·31-s + (−0.0170 + 0.0170i)35-s + (0.274 − 0.274i)37-s + 0.694i·41-s + (−0.528 − 0.528i)43-s + 0.229i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1337108143\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1337108143\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.909 - 0.909i)T - 25iT^{2} \) |
| 7 | \( 1 - 0.654T + 49T^{2} \) |
| 11 | \( 1 + (13.3 + 13.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (-8.32 - 8.32i)T + 169iT^{2} \) |
| 17 | \( 1 - 3.93T + 289T^{2} \) |
| 19 | \( 1 + (16.8 - 16.8i)T - 361iT^{2} \) |
| 23 | \( 1 + 23.1T + 529T^{2} \) |
| 29 | \( 1 + (35.6 + 35.6i)T + 841iT^{2} \) |
| 31 | \( 1 + 45.5iT - 961T^{2} \) |
| 37 | \( 1 + (-10.1 + 10.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 28.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (22.7 + 22.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 10.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (41.5 - 41.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (21.0 + 21.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (68.7 + 68.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (67.8 - 67.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 33.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 18.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 6.29iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (72.0 - 72.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 10.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−10.17880805499013993849606933646, −9.233589594660006029144533957549, −8.105420058255222482336540125505, −7.74101630222400779020347449147, −6.20814585964559742713092629242, −5.71371595906321156719042411844, −4.27406342553739340570989192657, −3.31124529317290738954087519364, −1.92837418465995175804034023601, −0.04742396593964375565217998868,
1.84095320519560966497208733082, 3.13141514325986579992180936933, 4.48850853964873346004474524720, 5.25994770621548781521330307529, 6.44491800419951910812623708053, 7.48668438925028028546689931037, 8.216203838455472867603617271534, 9.123779437065828400629054489748, 10.30906300943030757353393332340, 10.65635364197221057646778692200