L(s) = 1 | + (−0.766 + 0.642i)2-s + (2.78 + 1.01i)3-s + (0.173 − 0.984i)4-s + (−2.78 + 1.01i)6-s + (0.454 − 0.787i)7-s + (0.500 + 0.866i)8-s + (4.44 + 3.72i)9-s + (−1.82 − 3.16i)11-s + (1.48 − 2.56i)12-s + (2.66 − 0.969i)13-s + (0.157 + 0.895i)14-s + (−0.939 − 0.342i)16-s + (4.13 − 3.46i)17-s − 5.80·18-s + (3.41 + 2.70i)19-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (1.60 + 0.585i)3-s + (0.0868 − 0.492i)4-s + (−1.13 + 0.414i)6-s + (0.171 − 0.297i)7-s + (0.176 + 0.306i)8-s + (1.48 + 1.24i)9-s + (−0.550 − 0.953i)11-s + (0.428 − 0.741i)12-s + (0.738 − 0.268i)13-s + (0.0422 + 0.239i)14-s + (−0.234 − 0.0855i)16-s + (1.00 − 0.840i)17-s − 1.36·18-s + (0.783 + 0.621i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.10057 + 0.779427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10057 + 0.779427i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.41 - 2.70i)T \) |
good | 3 | \( 1 + (-2.78 - 1.01i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-0.454 + 0.787i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.82 + 3.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.66 + 0.969i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 3.46i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.226 + 1.28i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.16 - 2.65i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.98 - 5.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 + (0.0389 + 0.0141i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.54 - 8.74i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.74 - 1.46i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.76 + 10.0i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.30 + 3.61i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.646 - 3.66i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.4 + 8.80i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.01 - 5.75i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.2 + 4.81i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (14.0 + 5.12i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.39 - 4.15i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.41 - 3.06i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-12.7 + 10.7i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.07141294684181082058643185638, −9.031080812845158955762950656441, −8.550259536733526592079727927211, −7.83444131146426462250756114682, −7.19560010403224000538126895919, −5.79662946113432483482474781423, −4.85152881578050831671928888127, −3.50323548129631730148986788383, −2.95367657431335127307576819518, −1.34110062495276712785930962160,
1.42623431235149342819236205965, 2.29915436051297356058315475688, 3.25613474382515821054547695915, 4.16569888942459288790784014394, 5.68299075395518133880042007724, 7.17532433309578674705441601819, 7.49488360415455379249045403504, 8.518853700005796165486186835302, 8.873213389310837981511260120331, 9.844292003211635030033660554670