L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.483 − 0.837i)5-s + 0.999i·8-s + 0.967i·10-s + (4.82 − 2.78i)11-s + (3.76 + 2.17i)13-s + (−0.5 − 0.866i)16-s − 3.94·17-s + 4.46i·19-s + (−0.483 − 0.837i)20-s + (−2.78 + 4.82i)22-s + (2.29 + 1.32i)23-s + (2.03 + 3.51i)25-s − 4.35·26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.216 − 0.374i)5-s + 0.353i·8-s + 0.305i·10-s + (1.45 − 0.840i)11-s + (1.04 + 0.603i)13-s + (−0.125 − 0.216i)16-s − 0.956·17-s + 1.02i·19-s + (−0.108 − 0.187i)20-s + (−0.594 + 1.02i)22-s + (0.479 + 0.276i)23-s + (0.406 + 0.703i)25-s − 0.853·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.577109982\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.577109982\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.483 + 0.837i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.82 + 2.78i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.76 - 2.17i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 - 4.46iT - 19T^{2} \) |
| 23 | \( 1 + (-2.29 - 1.32i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.61 - 2.66i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.34 - 3.08i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.487T + 37T^{2} \) |
| 41 | \( 1 + (-0.0818 + 0.141i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.35 + 7.53i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.74 - 8.21i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.01iT - 53T^{2} \) |
| 59 | \( 1 + (0.836 - 1.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.47 + 2.58i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.72 + 4.71i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.64iT - 71T^{2} \) |
| 73 | \( 1 - 2.48iT - 73T^{2} \) |
| 79 | \( 1 + (2.30 + 3.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.20 + 7.29i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + (10.2 - 5.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.945860758366403359614615016558, −8.461025241147697184463483847359, −7.40337844877818056845936259492, −6.53856781567806693551104337250, −6.13125850119230750146442302476, −5.20122801534772077551090756024, −4.10077212524800628547526241595, −3.34870204142667223034912338983, −1.77589114636096579321978783626, −1.06620891177972077664673349767,
0.796012210570668233043923333265, 1.95014045674093051703094826664, 2.86704176691885879908409308412, 3.93351511280556464914050310201, 4.62779741741639665633077507230, 5.95467403768284939664343737997, 6.71035444402408240127693295020, 7.10816553174847046531849795405, 8.318487077513941267190129074172, 8.781142012574610642114549393738