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
−8.781142012574610642114549393738, −8.318487077513941267190129074172, −7.10816553174847046531849795405, −6.71035444402408240127693295020, −5.95467403768284939664343737997, −4.62779741741639665633077507230, −3.93351511280556464914050310201, −2.86704176691885879908409308412, −1.95014045674093051703094826664, −0.796012210570668233043923333265,
1.06620891177972077664673349767, 1.77589114636096579321978783626, 3.34870204142667223034912338983, 4.10077212524800628547526241595, 5.20122801534772077551090756024, 6.13125850119230750146442302476, 6.53856781567806693551104337250, 7.40337844877818056845936259492, 8.461025241147697184463483847359, 8.945860758366403359614615016558