| L(s) = 1 | + 2.82i·2-s − 8.00·4-s + 25.7i·5-s − 22.6i·8-s − 72.9·10-s − 2.36i·11-s + 268.·13-s + 64.0·16-s − 258. i·17-s − 527.·19-s − 206. i·20-s + 6.69·22-s + 128. i·23-s − 39.7·25-s + 759. i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.03i·5-s − 0.353i·8-s − 0.729·10-s − 0.0195i·11-s + 1.58·13-s + 0.250·16-s − 0.894i·17-s − 1.46·19-s − 0.515i·20-s + 0.0138·22-s + 0.242i·23-s − 0.0636·25-s + 1.12i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.909567691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909567691\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 25.7iT - 625T^{2} \) |
| 11 | \( 1 + 2.36iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 268.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 258. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 527.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 128. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.45e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 792.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.00e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.01e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.49e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 655. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.82e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.91e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.24e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.69e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 939.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.39e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.12e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.97e3T + 8.85e7T^{2} \) |
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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
−9.842753588379805060474126774628, −8.880269290948133604318858366681, −8.205130058024745326004761864779, −7.17404251969554649461770174454, −6.50984684400095286273251931039, −5.86000479186362472051740624065, −4.62524286541615983480000630244, −3.62191071003183966403356932547, −2.62698395331407491989697069808, −1.05075618878620863892465674694,
0.51114089002528717803868870465, 1.40620790200622108960038912435, 2.53037644435722690927428092719, 4.03888811044347503831221331037, 4.34980291002187553236970720767, 5.74552853128181617248803590915, 6.35248799277622495190042249877, 7.971400713443902104154176052296, 8.529265490793840862736309784835, 9.132996322746183127629317612620
