L(s) = 1 | + (0.918 − 1.07i)2-s − 1.19·3-s + (−0.313 − 1.97i)4-s + (−2.22 + 0.234i)5-s + (−1.09 + 1.28i)6-s + (2.11 − 1.58i)7-s + (−2.41 − 1.47i)8-s − 1.58·9-s + (−1.78 + 2.60i)10-s − 3.65·11-s + (0.373 + 2.35i)12-s − 5.56i·13-s + (0.239 − 3.73i)14-s + (2.64 − 0.279i)15-s + (−3.80 + 1.23i)16-s + 0.808·17-s + ⋯ |
L(s) = 1 | + (0.649 − 0.760i)2-s − 0.687·3-s + (−0.156 − 0.987i)4-s + (−0.994 + 0.105i)5-s + (−0.446 + 0.522i)6-s + (0.800 − 0.599i)7-s + (−0.852 − 0.521i)8-s − 0.527·9-s + (−0.565 + 0.824i)10-s − 1.10·11-s + (0.107 + 0.678i)12-s − 1.54i·13-s + (0.0639 − 0.997i)14-s + (0.683 − 0.0722i)15-s + (−0.950 + 0.309i)16-s + 0.196·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0832180 - 0.834552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0832180 - 0.834552i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.918 + 1.07i)T \) |
| 5 | \( 1 + (2.22 - 0.234i)T \) |
| 7 | \( 1 + (-2.11 + 1.58i)T \) |
good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 5.56iT - 13T^{2} \) |
| 17 | \( 1 - 0.808T + 17T^{2} \) |
| 19 | \( 1 - 4.54iT - 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 8.36iT - 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + 7.65iT - 41T^{2} \) |
| 43 | \( 1 - 2.66iT - 43T^{2} \) |
| 47 | \( 1 + 4.79iT - 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 - 8.92iT - 59T^{2} \) |
| 61 | \( 1 + 4.08T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 8.17iT - 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.49iT - 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 - 4.95iT - 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−11.53272271200591615650770960599, −10.59117413045543598196957332206, −10.25051482906816863337633585717, −8.305316577261994333020129948440, −7.65720677451510705528107992526, −5.98013680054806286491980756756, −5.17184428829571200665291088325, −4.09350261862954828159080598480, −2.80869287692501988264586290937, −0.55305571799106485877082923509,
2.83426798840242506191419538361, 4.52479104407922365953079393438, 5.07801599219743615846176546831, 6.27250212616319020382973390734, 7.34482652795576353352486158269, 8.284132675214082146270607240626, 9.023631292110468669105158752507, 10.95200043380768012214669228171, 11.58798952656683367928377432538, 12.12848373867896770450682011292