L(s) = 1 | + (0.5 + 0.866i)3-s + (1 − 1.73i)4-s + 2·5-s + (1.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 1.99·12-s + (−2.5 − 2.59i)13-s + (1 + 1.73i)15-s + (−1.99 − 3.46i)16-s + (−1 + 1.73i)17-s + (−2 + 3.46i)19-s + (2 − 3.46i)20-s + 3·21-s + (2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.5 − 0.866i)4-s + 0.894·5-s + (0.566 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.150 − 0.261i)11-s + 0.577·12-s + (−0.693 − 0.720i)13-s + (0.258 + 0.447i)15-s + (−0.499 − 0.866i)16-s + (−0.242 + 0.420i)17-s + (−0.458 + 0.794i)19-s + (0.447 − 0.774i)20-s + 0.654·21-s + (0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86650 - 0.512976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86650 - 0.512976i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.81678044773936391783532693265, −10.09843550297241741908486596020, −9.736898680263919613453495749469, −8.350052317819501847877473493045, −7.39087660368145928505961148274, −6.20740050845911403855201351866, −5.37661663680503849112181035886, −4.34252460016639136631536518347, −2.73462605109970399686229387303, −1.39720427620491547902621904285,
2.16678366987858238140260750155, 2.55122776933982552135848798009, 4.39589301572929641217737349104, 5.64539120740918972859028818568, 6.73397557999703182995456990489, 7.45289854872019843962536436685, 8.660869948825948615679052357405, 9.086959780318959595116549245163, 10.36546363481075315041401481862, 11.56255950723112049947592506413