L(s) = 1 | + (0.5 − 0.866i)3-s + (2 + 3.46i)5-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s − 4·13-s + 3.99·15-s + (−2 − 3.46i)19-s + (−5.49 + 9.52i)25-s − 0.999·27-s + 2·29-s + (−4 + 6.92i)31-s + (1.99 + 3.46i)33-s + (3 + 5.19i)37-s + (−2 + 3.46i)39-s − 4·43-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.894 + 1.54i)5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s − 1.10·13-s + 1.03·15-s + (−0.458 − 0.794i)19-s + (−1.09 + 1.90i)25-s − 0.192·27-s + 0.371·29-s + (−0.718 + 1.24i)31-s + (0.348 + 0.603i)33-s + (0.493 + 0.854i)37-s + (−0.320 + 0.554i)39-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.371647335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371647335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.453430356924096516954387966619, −8.426398773470326105187663785078, −7.30748904774936482182526723639, −7.11932411701363410314260889340, −6.38403530444109867794797676531, −5.42904588324308453994154679732, −4.53158274287161315924384974500, −3.09344413618960093487417649904, −2.55339438013046948776458137178, −1.78714223610492468647488663261,
0.40047566782599084506419003243, 1.81074449269045387743393060542, 2.73251872007533968775631724752, 4.02994866747904851080724044708, 4.74452464209366695153229308184, 5.63029002947119587336120004431, 5.90035329179264608648699729110, 7.38499561046001567850267299425, 8.204943654080108318845110275604, 8.774744769934464720347236597420