| L(s) = 1 | + (−0.156 − 0.987i)2-s + (−0.833 − 0.424i)3-s + (−0.951 + 0.309i)4-s + (−2.09 + 0.774i)5-s + (−0.289 + 0.889i)6-s + (2.58 − 2.58i)7-s + (0.453 + 0.891i)8-s + (−1.24 − 1.71i)9-s + (1.09 + 1.95i)10-s + (3.46 + 2.51i)11-s + (0.924 + 0.146i)12-s + (0.601 − 3.79i)13-s + (−2.95 − 2.14i)14-s + (2.07 + 0.245i)15-s + (0.809 − 0.587i)16-s + (2.28 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (−0.481 − 0.245i)3-s + (−0.475 + 0.154i)4-s + (−0.938 + 0.346i)5-s + (−0.118 + 0.363i)6-s + (0.975 − 0.975i)7-s + (0.160 + 0.315i)8-s + (−0.416 − 0.572i)9-s + (0.345 + 0.616i)10-s + (1.04 + 0.758i)11-s + (0.266 + 0.0422i)12-s + (0.166 − 1.05i)13-s + (−0.789 − 0.573i)14-s + (0.536 + 0.0633i)15-s + (0.202 − 0.146i)16-s + (0.553 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0833168 - 0.781309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0833168 - 0.781309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 + (2.09 - 0.774i)T \) |
| 19 | \( 1 + (1.76 + 3.98i)T \) |
| good | 3 | \( 1 + (0.833 + 0.424i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-2.58 + 2.58i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.46 - 2.51i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.601 + 3.79i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.28 - 4.47i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-0.419 - 2.64i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.92 + 8.98i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.54 + 1.47i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.33 + 1.00i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.395 + 0.544i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.0431 + 0.0431i)T + 43iT^{2} \) |
| 47 | \( 1 + (10.4 + 5.31i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (4.61 + 2.35i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.3 + 7.49i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.46 + 4.69i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.99 - 2.54i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (8.49 - 2.75i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0348 - 0.219i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.951 - 2.92i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.33 + 3.22i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.123 - 0.0896i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.22 + 12.2i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−9.905457224121902366506220102907, −8.773979289121780371719694461935, −7.943892140618772424870485129215, −7.27266393619986262413583430341, −6.29575126091426641502970533429, −5.04582814923461509106558211630, −4.01699572495885029446796608697, −3.44620267625916156528128243041, −1.68291890480722994473047398282, −0.44110682418023755188855037841,
1.54276442771112619552120203081, 3.40437404881330704170998560848, 4.58436020761284038670766492351, 5.18338374760749677544746652937, 6.03973957779472149122429516277, 7.09189499820847260484938880975, 8.017339558521578571519042007546, 8.784216805761039993877848645614, 9.091293499249800456518391740566, 10.58247516532441984449736047777
