L(s) = 1 | + (−2.67 − 1.54i)2-s + (2.77 + 4.81i)4-s + (−1.93 + 1.11i)5-s + (1.10 − 1.91i)7-s − 4.80i·8-s + 6.91·10-s + (15.4 + 8.91i)11-s + (1.25 + 2.17i)13-s + (−5.90 + 3.40i)14-s + (3.67 − 6.37i)16-s − 32.6i·17-s + 7.93·19-s + (−10.7 − 6.21i)20-s + (−27.5 − 47.7i)22-s + (18.4 − 10.6i)23-s + ⋯ |
L(s) = 1 | + (−1.33 − 0.772i)2-s + (0.694 + 1.20i)4-s + (−0.387 + 0.223i)5-s + (0.157 − 0.272i)7-s − 0.601i·8-s + 0.691·10-s + (1.40 + 0.810i)11-s + (0.0966 + 0.167i)13-s + (−0.421 + 0.243i)14-s + (0.229 − 0.398i)16-s − 1.91i·17-s + 0.417·19-s + (−0.537 − 0.310i)20-s + (−1.25 − 2.17i)22-s + (0.801 − 0.462i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.667577 - 0.312212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.667577 - 0.312212i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
good | 2 | \( 1 + (2.67 + 1.54i)T + (2 + 3.46i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 1.91i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.4 - 8.91i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 2.17i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 32.6iT - 289T^{2} \) |
| 19 | \( 1 - 7.93T + 361T^{2} \) |
| 23 | \( 1 + (-18.4 + 10.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-30.7 - 17.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (1.01 + 1.75i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 50.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (4.65 - 2.68i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (7.76 - 13.4i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.63 - 0.943i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 62.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (31.3 - 18.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21.1 + 36.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.38 - 12.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 105. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 66.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-34.5 + 59.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-18.3 - 10.6i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 7.16iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (55.6 - 96.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.24308884250389095265470887743, −11.64001101441626395453714325693, −10.77299038870374082170521543502, −9.582708361000478912467076262798, −9.038508944480640100516859677214, −7.66547060457785130621371210550, −6.82231098836579554973960320206, −4.61095301618316541354628781693, −2.86781800405798288038098654520, −1.06649051308553459756556039397,
1.15992521836346367086397223227, 3.84733502381457299361656610155, 5.85736325773102479074492312927, 6.79223263605157096053545588210, 8.149442850449227355276556709364, 8.674821462870110040203771338924, 9.698594728417999993347044496745, 10.87029682953633922900201312607, 11.86996411513131577095339769601, 13.13393050432509137149133456644