L(s) = 1 | − i·2-s + (1.33 + 1.10i)3-s − 4-s + (1.94 + 3.36i)5-s + (1.10 − 1.33i)6-s + i·8-s + (0.577 + 2.94i)9-s + (3.36 − 1.94i)10-s + (−3.41 − 1.97i)11-s + (−1.33 − 1.10i)12-s + (2.46 + 1.42i)13-s + (−1.10 + 6.64i)15-s + 16-s + (−0.371 − 0.642i)17-s + (2.94 − 0.577i)18-s + (1.54 + 0.892i)19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.772 + 0.635i)3-s − 0.5·4-s + (0.870 + 1.50i)5-s + (0.449 − 0.546i)6-s + 0.353i·8-s + (0.192 + 0.981i)9-s + (1.06 − 0.615i)10-s + (−1.03 − 0.594i)11-s + (−0.386 − 0.317i)12-s + (0.684 + 0.395i)13-s + (−0.285 + 1.71i)15-s + 0.250·16-s + (−0.0899 − 0.155i)17-s + (0.693 − 0.136i)18-s + (0.354 + 0.204i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88470 + 0.975601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88470 + 0.975601i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.33 - 1.10i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.41 + 1.97i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.371 + 0.642i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 0.892i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.41 + 3.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.50 - 1.44i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.51iT - 31T^{2} \) |
| 37 | \( 1 + (1.50 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.24 - 9.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 0.0211T + 59T^{2} \) |
| 61 | \( 1 + 2.46iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 1.94iT - 71T^{2} \) |
| 73 | \( 1 + (-4.20 + 2.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 + (4.02 + 6.98i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.2 + 9.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.22818791195989970387779136816, −9.713133492530123730869281849949, −8.802885087598830432215309537623, −7.920756927483503347601123071469, −6.86484627977464331543801647211, −5.81711209606824803758356920134, −4.79870902171144893244323879375, −3.39331853410552546180700702295, −2.93739617380118159285072520449, −1.94628629956668636247032242094,
0.981096951439944518196630057216, 2.17523691815868291847584730992, 3.67362851023365637818965673565, 5.03762117114525627495438021215, 5.51139372885655646166685619555, 6.65497382340941102920509056603, 7.60351881780987224530947794474, 8.342258158888782573265160717691, 9.011767242869126846046517648930, 9.567155484379644325234715580045