L(s) = 1 | + 2.51i·2-s − i·3-s − 4.34·4-s + (−1.27 − 1.83i)5-s + 2.51·6-s + 0.173i·7-s − 5.90i·8-s − 9-s + (4.62 − 3.20i)10-s + 5.08·11-s + 4.34i·12-s − 1.82i·13-s − 0.436·14-s + (−1.83 + 1.27i)15-s + 6.19·16-s − 4.24i·17-s + ⋯ |
L(s) = 1 | + 1.78i·2-s − 0.577i·3-s − 2.17·4-s + (−0.569 − 0.821i)5-s + 1.02·6-s + 0.0655i·7-s − 2.08i·8-s − 0.333·9-s + (1.46 − 1.01i)10-s + 1.53·11-s + 1.25i·12-s − 0.506i·13-s − 0.116·14-s + (−0.474 + 0.328i)15-s + 1.54·16-s − 1.02i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07073 + 0.334901i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07073 + 0.334901i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.27 + 1.83i)T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.51iT - 2T^{2} \) |
| 7 | \( 1 - 0.173iT - 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 + 1.82iT - 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 8.62T + 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 31 | \( 1 - 3.22T + 31T^{2} \) |
| 37 | \( 1 - 1.97iT - 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 + 7.91iT - 43T^{2} \) |
| 47 | \( 1 + 8.66iT - 47T^{2} \) |
| 53 | \( 1 + 5.40iT - 53T^{2} \) |
| 59 | \( 1 + 7.66T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 8.13iT - 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 - 6.74iT - 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 2.74iT - 97T^{2} \) |
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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
−11.72499743842598479169961935383, −9.775905051048762296206552857606, −8.954366826204978780472727740872, −8.331869401286255530983125415286, −7.34165493705698315603096436168, −6.82571383342698626744473048381, −5.61664760318209927021092219972, −4.89223943211723950137535025966, −3.63512508502656866723192111736, −0.851196706449243457577737574702,
1.46681786629662789874171379008, 3.12367358760084048251845704331, 3.73393973915889489529018765408, 4.62419097202977407753476878517, 6.19962614103983441353820108384, 7.57733197038064211091659495233, 8.884421099779445911222063760602, 9.564852368127007324699239570163, 10.30511666117055469018392475445, 11.20870354285811323313436612716