L(s) = 1 | + (6.08 + 6.08i)5-s + 9.40i·7-s + (−11.9 − 11.9i)11-s + (−3.47 − 3.47i)13-s + 28.5i·17-s + (3.08 + 3.08i)19-s + 2.57·23-s + 49.1i·25-s + (−3.49 + 3.49i)29-s − 21.0·31-s + (−57.2 + 57.2i)35-s + (13.2 − 13.2i)37-s − 11.2·41-s + (−8.19 + 8.19i)43-s + 17.2i·47-s + ⋯ |
L(s) = 1 | + (1.21 + 1.21i)5-s + 1.34i·7-s + (−1.08 − 1.08i)11-s + (−0.267 − 0.267i)13-s + 1.67i·17-s + (0.162 + 0.162i)19-s + 0.111·23-s + 1.96i·25-s + (−0.120 + 0.120i)29-s − 0.680·31-s + (−1.63 + 1.63i)35-s + (0.359 − 0.359i)37-s − 0.274·41-s + (−0.190 + 0.190i)43-s + 0.366i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.608563128\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608563128\) |
\(L(2)\) |
|
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 | \( 1 + (-6.08 - 6.08i)T + 25iT^{2} \) |
| 7 | \( 1 - 9.40iT - 49T^{2} \) |
| 11 | \( 1 + (11.9 + 11.9i)T + 121iT^{2} \) |
| 13 | \( 1 + (3.47 + 3.47i)T + 169iT^{2} \) |
| 17 | \( 1 - 28.5iT - 289T^{2} \) |
| 19 | \( 1 + (-3.08 - 3.08i)T + 361iT^{2} \) |
| 23 | \( 1 - 2.57T + 529T^{2} \) |
| 29 | \( 1 + (3.49 - 3.49i)T - 841iT^{2} \) |
| 31 | \( 1 + 21.0T + 961T^{2} \) |
| 37 | \( 1 + (-13.2 + 13.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 11.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (8.19 - 8.19i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 17.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-30.5 - 30.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (14.7 + 14.7i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (39.3 + 39.3i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (79.6 + 79.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 73.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 1.23iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-13.7 + 13.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 39.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−10.05915533180828312951584762666, −9.158112146582503412066626340878, −8.389613639672232485697770665135, −7.49614855055207700795458923495, −6.18994695490384627181313987900, −5.94688419441738378236633289338, −5.19661895605407953096522359177, −3.41035829370138544308338032922, −2.66415954380569835409271237753, −1.85163469868176904473443044885,
0.44417111622188834220835302126, 1.60792703032851118225621969604, 2.69705688469470746159282577122, 4.33796748810195097113910485791, 4.92600889993529403745117065304, 5.62046912548155907840204608904, 6.99804301904168506631867632856, 7.42828500934091649509045725619, 8.549502957896154711118794355725, 9.533559948443352148548191627892