L(s) = 1 | + (1.13 + 1.97i)2-s + (1.41 − 2.44i)4-s + (−2.27 − 3.94i)5-s + 24.6·8-s + (5.17 − 8.96i)10-s + (−20.3 + 35.2i)11-s + 53.2·13-s + (16.7 + 28.9i)16-s + (2.27 − 3.94i)17-s + (−61.2 − 106. i)19-s − 12.8·20-s − 92.7·22-s + (65.6 + 113. i)23-s + (52.1 − 90.3i)25-s + (60.6 + 105. i)26-s + ⋯ |
L(s) = 1 | + (0.402 + 0.696i)2-s + (0.176 − 0.305i)4-s + (−0.203 − 0.352i)5-s + 1.08·8-s + (0.163 − 0.283i)10-s + (−0.558 + 0.967i)11-s + 1.13·13-s + (0.261 + 0.452i)16-s + (0.0324 − 0.0562i)17-s + (−0.740 − 1.28i)19-s − 0.143·20-s − 0.898·22-s + (0.595 + 1.03i)23-s + (0.417 − 0.722i)25-s + (0.457 + 0.792i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.737745716\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.737745716\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.13 - 1.97i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (2.27 + 3.94i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (20.3 - 35.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.27 + 3.94i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (61.2 + 106. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-65.6 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-125. + 218. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (5.94 + 10.3i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (131. + 227. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (283. - 491. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-419. + 727. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-242. - 420. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-166. + 288. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (245. - 424. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (60.8 + 105. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-359. - 622. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
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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
−10.75702013305947847779750788496, −9.872094858744564351142329457913, −8.754342331719304119437162784433, −7.79457273251849276839566237018, −6.88217638156899595831087168303, −6.05575762583689036866682641091, −4.94652375154764174859598021832, −4.25058211168099495238479215315, −2.46499507995740152647152630667, −0.952161377659516411008576707828,
1.19314018676567476159778557209, 2.74207287823686732510167972984, 3.50068073048632801504316173733, 4.57679872110058763843848078805, 5.95781937289982003910301433886, 6.91160243297082169381731331586, 8.150429222369500526780073658588, 8.620926516051447965672343158808, 10.33050253355976972248021500549, 10.75727860500200315673724851696