L(s) = 1 | + (12.2 − 29.5i)2-s + (−126. + 126. i)3-s + (−722. − 725. i)4-s + (−1.65e3 + 1.65e3i)5-s + (2.17e3 + 5.27e3i)6-s + 2.98e4·7-s + (−3.03e4 + 1.24e4i)8-s + 2.72e4i·9-s + (2.85e4 + 6.92e4i)10-s + (8.99e4 + 8.99e4i)11-s + (1.82e5 + 391. i)12-s + (5.66e3 + 5.66e3i)13-s + (3.66e5 − 8.82e5i)14-s − 4.17e5i·15-s + (−4.49e3 + 1.04e6i)16-s + 2.33e6·17-s + ⋯ |
L(s) = 1 | + (0.383 − 0.923i)2-s + (−0.518 + 0.518i)3-s + (−0.705 − 0.708i)4-s + (−0.529 + 0.529i)5-s + (0.280 + 0.678i)6-s + 1.77·7-s + (−0.925 + 0.379i)8-s + 0.461i·9-s + (0.285 + 0.692i)10-s + (0.558 + 0.558i)11-s + (0.733 + 0.00157i)12-s + (0.0152 + 0.0152i)13-s + (0.681 − 1.64i)14-s − 0.549i·15-s + (−0.00428 + 0.999i)16-s + 1.64·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.59830 + 0.314361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59830 + 0.314361i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-12.2 + 29.5i)T \) |
good | 3 | \( 1 + (126. - 126. i)T - 5.90e4iT^{2} \) |
| 5 | \( 1 + (1.65e3 - 1.65e3i)T - 9.76e6iT^{2} \) |
| 7 | \( 1 - 2.98e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + (-8.99e4 - 8.99e4i)T + 2.59e10iT^{2} \) |
| 13 | \( 1 + (-5.66e3 - 5.66e3i)T + 1.37e11iT^{2} \) |
| 17 | \( 1 - 2.33e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + (2.56e6 - 2.56e6i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 + 2.91e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + (4.43e6 + 4.43e6i)T + 4.20e14iT^{2} \) |
| 31 | \( 1 - 2.94e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + (-1.06e7 + 1.06e7i)T - 4.80e15iT^{2} \) |
| 41 | \( 1 - 5.47e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + (-7.25e7 - 7.25e7i)T + 2.16e16iT^{2} \) |
| 47 | \( 1 + 2.46e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + (-3.94e8 + 3.94e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + (7.21e8 + 7.21e8i)T + 5.11e17iT^{2} \) |
| 61 | \( 1 + (-5.12e8 - 5.12e8i)T + 7.13e17iT^{2} \) |
| 67 | \( 1 + (-1.09e8 + 1.09e8i)T - 1.82e18iT^{2} \) |
| 71 | \( 1 + 1.01e7T + 3.25e18T^{2} \) |
| 73 | \( 1 + 2.60e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 2.96e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (3.36e9 - 3.36e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 2.18e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.07e10T + 7.37e19T^{2} \) |
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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
−16.94202869730406230534926121321, −14.94764217932204156470633033459, −14.26154654111142297067479123723, −12.12432708166634984202174279813, −11.20268089314098821907250947123, −10.21213782421414154488979286009, −8.038657744551142711967789038187, −5.32491600324809959614807033077, −4.04409625012060438103233046402, −1.67794473160354989458554801302,
0.815337656967721325875680969512, 4.28620947640305513464650068745, 5.78309879958459949354061416544, 7.55939872788550170365207408561, 8.690236769516716376366818958602, 11.50396064342120688338941009530, 12.42400862871910608625357905248, 14.13020747353196360169098526106, 15.14721198277564454139624920932, 16.77565298471465540084489217964