| L(s) = 1 | + 5.38i·2-s − 12.9·4-s + 37.0i·5-s − 57.6·7-s + 16.4i·8-s − 199.·10-s + 53.4i·11-s − 303.·13-s − 310. i·14-s − 295.·16-s + 552. i·17-s + 321.·19-s − 479. i·20-s − 287.·22-s − 231. i·23-s + ⋯ |
| L(s) = 1 | + 1.34i·2-s − 0.809·4-s + 1.48i·5-s − 1.17·7-s + 0.256i·8-s − 1.99·10-s + 0.441i·11-s − 1.79·13-s − 1.58i·14-s − 1.15·16-s + 1.91i·17-s + 0.889·19-s − 1.19i·20-s − 0.594·22-s − 0.437i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7972119418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7972119418\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 - 5.38iT - 16T^{2} \) |
| 5 | \( 1 - 37.0iT - 625T^{2} \) |
| 7 | \( 1 + 57.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 53.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 303.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 552. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 321.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 231. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 833. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.69e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 921.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 951. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.10e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.12e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.20e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 7.31e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.45e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.92e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.30e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.06e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.67e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.20e4T + 8.85e7T^{2} \) |
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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
−10.69197999109829190327706266574, −10.12506279891508337904416181890, −9.229818300428778331771095426672, −7.924276287209355501946877702681, −7.22321679666117913776696048130, −6.58767419733160170043686441582, −5.98830433899868932877737902224, −4.71934933402386029724252238320, −3.29188224820002989989673376368, −2.34413074776459540637499985264,
0.25831924272943528284405749959, 0.849633353992161238371071112861, 2.41793562978111449243996816842, 3.23070002697245930747772703996, 4.55695351992052005792679950815, 5.24612021721645713945879415279, 6.71442054465367990868497521573, 7.77126496141934100184271883221, 9.138635111170000089651151422058, 9.613778939395453402215082979366
