L(s) = 1 | − 2.05·2-s + 2.22·4-s + (0.274 + 0.475i)5-s + (2.59 + 0.527i)7-s − 0.456·8-s + (−0.564 − 0.977i)10-s + (2.34 + 4.06i)11-s + (−0.663 + 3.54i)13-s + (−5.32 − 1.08i)14-s − 3.50·16-s + 0.603·17-s + (−0.280 + 0.485i)19-s + (0.610 + 1.05i)20-s + (−4.82 − 8.36i)22-s − 0.376·23-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.11·4-s + (0.122 + 0.212i)5-s + (0.979 + 0.199i)7-s − 0.161·8-s + (−0.178 − 0.309i)10-s + (0.708 + 1.22i)11-s + (−0.184 + 0.982i)13-s + (−1.42 − 0.289i)14-s − 0.876·16-s + 0.146·17-s + (−0.0643 + 0.111i)19-s + (0.136 + 0.236i)20-s + (−1.02 − 1.78i)22-s − 0.0785·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.608787 + 0.500913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.608787 + 0.500913i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.59 - 0.527i)T \) |
| 13 | \( 1 + (0.663 - 3.54i)T \) |
good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 5 | \( 1 + (-0.274 - 0.475i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.34 - 4.06i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.603T + 17T^{2} \) |
| 19 | \( 1 + (0.280 - 0.485i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.376T + 23T^{2} \) |
| 29 | \( 1 + (2.09 - 3.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.577 - 0.999i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.80T + 37T^{2} \) |
| 41 | \( 1 + (-3.96 + 6.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.747 + 1.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.09 - 1.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.52 - 7.83i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 + (3.71 - 6.42i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.79 - 8.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.88 + 5.00i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.24 + 12.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 + (-3.15 - 5.45i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.40611551986537825605527898116, −9.260604413055936480911718628088, −9.029133896822089046861100163689, −7.944567735902476790229084587750, −7.22179429428332717406291982358, −6.53060925092329530621096650796, −5.02462068128003135740323245810, −4.14470013810080095570389025773, −2.22672624874443699124917649464, −1.46196051387890084140358705553,
0.69387563448315584187069773466, 1.77425460781925995927203942956, 3.34777076588191056755366528579, 4.74571729278987261258452817960, 5.77597453792977817077378175429, 6.91340316361758340174533597549, 7.904152253471330272825906701630, 8.329293801820155141710654808899, 9.143244510409603830257443989790, 9.884551346328273171828306328434