| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (1.48 − 2.56i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (−1.71 − 2.97i)11-s + (−0.499 + 0.866i)12-s + (−1.25 − 1.05i)13-s + (2.78 + 1.01i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.0455 − 0.258i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (0.312 + 0.262i)6-s + (0.560 − 0.971i)7-s + (−0.176 − 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (−0.517 − 0.897i)11-s + (−0.144 + 0.249i)12-s + (−0.348 − 0.292i)13-s + (0.745 + 0.271i)14-s + (−0.242 + 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.0110 − 0.0626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32432 - 0.594165i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32432 - 0.594165i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (0.0583 + 4.35i)T \) |
| good | 7 | \( 1 + (-1.48 + 2.56i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.71 + 2.97i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.25 + 1.05i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0455 + 0.258i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.142 - 0.0517i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.841 - 4.77i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.71 + 2.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + (-4.43 + 3.72i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.567 - 0.206i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.582 + 3.30i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (11.1 - 4.05i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.473 - 2.68i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.41 - 0.879i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.0888 + 0.503i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 1.58i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.88 - 4.10i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.61 + 3.86i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.78 - 6.55i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.99 - 5.86i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.28 + 7.27i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.72864634402428923475666243547, −9.512411768782078591102119732691, −8.581109223849221594453174819691, −7.76998269030209447337560009296, −7.31656201022357867461155222079, −6.17300460421642346942929970597, −4.99308901877734087307556792754, −4.06706471543160694442064294311, −2.83627498013161821591665412058, −0.77907685047307613362082431822,
1.94154478697285314420733437077, 2.87435695189639144718925890632, 4.20627585742640751656585244031, 4.94800274335283705832601171787, 6.11067679150132814643643875066, 7.67169491308396029913880318843, 8.221256217141243062983236196079, 9.337885750438324981588790161603, 9.915871921708108084849780096808, 10.91285934798996389608281554818
