L(s) = 1 | + (−1.49 + 0.400i)2-s − i·3-s + (0.345 − 0.199i)4-s + (0.481 + 0.128i)5-s + (0.400 + 1.49i)6-s + (−2.58 + 0.560i)7-s + (1.75 − 1.75i)8-s − 9-s − 0.771·10-s + (1.32 − 1.32i)11-s + (−0.199 − 0.345i)12-s + (−3.48 − 0.924i)13-s + (3.64 − 1.87i)14-s + (0.128 − 0.481i)15-s + (−2.31 + 4.01i)16-s + (−1.95 − 3.39i)17-s + ⋯ |
L(s) = 1 | + (−1.05 + 0.283i)2-s − 0.577i·3-s + (0.172 − 0.0997i)4-s + (0.215 + 0.0576i)5-s + (0.163 + 0.610i)6-s + (−0.977 + 0.211i)7-s + (0.619 − 0.619i)8-s − 0.333·9-s − 0.244·10-s + (0.398 − 0.398i)11-s + (−0.0576 − 0.0997i)12-s + (−0.966 − 0.256i)13-s + (0.973 − 0.501i)14-s + (0.0333 − 0.124i)15-s + (−0.579 + 1.00i)16-s + (−0.475 − 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0721024 - 0.216411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0721024 - 0.216411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.58 - 0.560i)T \) |
| 13 | \( 1 + (3.48 + 0.924i)T \) |
good | 2 | \( 1 + (1.49 - 0.400i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.481 - 0.128i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.32 + 1.32i)T - 11iT^{2} \) |
| 17 | \( 1 + (1.95 + 3.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 - 2.70i)T - 19iT^{2} \) |
| 23 | \( 1 + (4.25 + 2.45i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.28 + 3.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.12 + 4.21i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.0389 - 0.145i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.92 - 1.31i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (5.06 + 2.92i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.39 - 8.92i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.56 + 11.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.986 - 3.68i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 11.7iT - 61T^{2} \) |
| 67 | \( 1 + (-6.99 - 6.99i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.82 - 0.487i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-6.55 + 1.75i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.22 - 3.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.85 - 5.85i)T - 83iT^{2} \) |
| 89 | \( 1 + (-11.9 + 3.20i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (2.09 + 7.81i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.51253195243645958216870548648, −10.11428980765982644724974839776, −9.627608727665333720299599651200, −8.624286394019547899811741704195, −7.72084075184551184637338243825, −6.77263431890529767440160088571, −5.88973830059059610396644413423, −4.07140453541014326592006500000, −2.34823330574284557651812719561, −0.23281341110576393899477193916,
2.07459250169217683611950014448, 3.83801901294048314887935300989, 5.08087185950562934676613081643, 6.49002760697873324252898542387, 7.62415438542603476472086591404, 8.886872306803321782469533369552, 9.485780338211687607896611744357, 10.15415050741617991988678885911, 10.93596074039960264731621366177, 12.06929168260183321357166250727