L(s) = 1 | + (−0.769 − 1.18i)2-s + (−0.808 + 0.808i)3-s + (−0.817 + 1.82i)4-s + (0.885 − 2.05i)5-s + (1.58 + 0.337i)6-s + (0.771 + 0.771i)7-s + (2.79 − 0.433i)8-s + 1.69i·9-s + (−3.11 + 0.527i)10-s + 0.875i·11-s + (−0.815 − 2.13i)12-s + (0.707 + 0.707i)13-s + (0.322 − 1.50i)14-s + (0.943 + 2.37i)15-s + (−2.66 − 2.98i)16-s + (4.91 − 4.91i)17-s + ⋯ |
L(s) = 1 | + (−0.543 − 0.839i)2-s + (−0.466 + 0.466i)3-s + (−0.408 + 0.912i)4-s + (0.396 − 0.918i)5-s + (0.645 + 0.137i)6-s + (0.291 + 0.291i)7-s + (0.988 − 0.153i)8-s + 0.563i·9-s + (−0.985 + 0.166i)10-s + 0.264i·11-s + (−0.235 − 0.616i)12-s + (0.196 + 0.196i)13-s + (0.0861 − 0.403i)14-s + (0.243 + 0.613i)15-s + (−0.666 − 0.745i)16-s + (1.19 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.882839 - 0.271397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.882839 - 0.271397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.769 + 1.18i)T \) |
| 5 | \( 1 + (-0.885 + 2.05i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.808 - 0.808i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.771 - 0.771i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.875iT - 11T^{2} \) |
| 17 | \( 1 + (-4.91 + 4.91i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + (-6.03 + 6.03i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 5.45iT - 31T^{2} \) |
| 37 | \( 1 + (2.21 - 2.21i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 + (4.34 - 4.34i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.81 - 3.81i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.18 + 2.18i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.749T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 + (10.0 + 10.0i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.01iT - 71T^{2} \) |
| 73 | \( 1 + (-6.85 - 6.85i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 + (2.66 - 2.66i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.68iT - 89T^{2} \) |
| 97 | \( 1 + (-6.65 + 6.65i)T - 97iT^{2} \) |
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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
−11.85457649526911388928019629633, −10.91099590132179039448759844020, −9.987777256682275476264108405381, −9.244660251027148776761790868367, −8.337622416150002605294118838492, −7.21132601200113862113260700599, −5.20161728537240796363734111809, −4.83936066700335034108873595491, −3.03680219669892756700343137296, −1.30068546197446560560855618037,
1.27827706634727027198069554552, 3.54934793031314028018232239969, 5.52963726309446924311493843582, 6.07258981934103798856638522829, 7.23147523389067527755793582625, 7.79292328484808263699437398626, 9.267733260837690643400131937773, 10.05229636665206974852074323271, 11.01498165960975678089548601192, 11.86782289142791261605021415541