| L(s) = 1 | + (0.146 + 1.40i)2-s + (0.0621 − 1.73i)3-s + (−1.95 + 0.413i)4-s + (3.40 − 1.58i)5-s + (2.44 − 0.166i)6-s + (−0.786 + 4.46i)7-s + (−0.869 − 2.69i)8-s + (−2.99 − 0.215i)9-s + (2.73 + 4.55i)10-s + (3.07 + 1.43i)11-s + (0.594 + 3.41i)12-s + (0.0327 − 0.373i)13-s + (−6.39 − 0.450i)14-s + (−2.53 − 5.99i)15-s + (3.65 − 1.61i)16-s + (2.48 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (0.103 + 0.994i)2-s + (0.0358 − 0.999i)3-s + (−0.978 + 0.206i)4-s + (1.52 − 0.709i)5-s + (0.997 − 0.0681i)6-s + (−0.297 + 1.68i)7-s + (−0.307 − 0.951i)8-s + (−0.997 − 0.0717i)9-s + (0.864 + 1.44i)10-s + (0.926 + 0.431i)11-s + (0.171 + 0.985i)12-s + (0.00907 − 0.103i)13-s + (−1.70 − 0.120i)14-s + (−0.654 − 1.54i)15-s + (0.914 − 0.404i)16-s + (0.603 − 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63565 + 0.466812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63565 + 0.466812i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.146 - 1.40i)T \) |
| 3 | \( 1 + (-0.0621 + 1.73i)T \) |
| good | 5 | \( 1 + (-3.40 + 1.58i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (0.786 - 4.46i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 1.43i)T + (7.07 + 8.42i)T^{2} \) |
| 13 | \( 1 + (-0.0327 + 0.373i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 1.43i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.54 + 1.75i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.58 + 0.455i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.395 + 4.52i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (9.47 - 1.67i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.275 - 1.02i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (3.38 - 2.84i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 4.52i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.03 - 5.88i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (4.41 - 4.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.21 - 2.61i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (5.01 + 7.16i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (1.09 + 0.0958i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (2.28 - 1.31i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.225 + 0.130i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.98 - 9.51i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.12 - 0.798i)T + (81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-1.56 + 2.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.783 + 0.284i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.67834867674018788471231452403, −9.622591904634462436891170562961, −9.274100002544426291056421010578, −8.619981937058072973253867357864, −7.37883757366507627060233881554, −6.36154849667929466652365937324, −5.65782507490386430721135745749, −5.15287630412177576144220284646, −2.91844194161173325324691431672, −1.48501414438738052794197562558,
1.42462136113788267413696849903, 3.21744724071735485552082486182, 3.74999926941458417111697352051, 5.15220401666947724689552615856, 6.05342654921686860595167942120, 7.33201624680279070959227016319, 8.978309023709091359119882254517, 9.658342229246368285891771510267, 10.21735758462587199844734167964, 10.80274215756608915205894496633
