| L(s) = 1 | + (0.868 + 1.11i)2-s + (−0.327 − 1.00i)3-s + (−0.491 + 1.93i)4-s + (−1.12 + 1.92i)5-s + (0.840 − 1.24i)6-s + 3.62i·7-s + (−2.59 + 1.13i)8-s + (1.51 − 1.10i)9-s + (−3.13 + 0.415i)10-s + (0.116 − 0.160i)11-s + (2.11 − 0.139i)12-s + (1.57 − 1.14i)13-s + (−4.04 + 3.14i)14-s + (2.31 + 0.506i)15-s + (−3.51 − 1.90i)16-s + (5.99 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (0.614 + 0.789i)2-s + (−0.189 − 0.582i)3-s + (−0.245 + 0.969i)4-s + (−0.505 + 0.863i)5-s + (0.343 − 0.506i)6-s + 1.36i·7-s + (−0.915 + 0.401i)8-s + (0.506 − 0.367i)9-s + (−0.991 + 0.131i)10-s + (0.0351 − 0.0483i)11-s + (0.610 − 0.0403i)12-s + (0.437 − 0.318i)13-s + (−1.08 + 0.840i)14-s + (0.597 + 0.130i)15-s + (−0.879 − 0.476i)16-s + (1.45 + 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.891256 + 1.05832i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891256 + 1.05832i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.868 - 1.11i)T \) |
| 5 | \( 1 + (1.12 - 1.92i)T \) |
| good | 3 | \( 1 + (0.327 + 1.00i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 3.62iT - 7T^{2} \) |
| 11 | \( 1 + (-0.116 + 0.160i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.57 + 1.14i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.99 - 1.94i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (3.31 + 1.07i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.186 - 0.256i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.98 + 0.644i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 7.39i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.14 + 2.28i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.48 - 1.80i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + (7.86 - 2.55i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.48 - 13.7i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.165 - 0.227i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 2.57i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.0384 + 0.118i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.42 + 4.38i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.51 + 10.3i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.42 - 4.37i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.15 - 12.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (13.8 + 10.0i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.24 - 0.730i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.54022332704486572284847176188, −12.18698445402166826644956767911, −11.14836943958518864893628203505, −9.612564468196691174445601282156, −8.298695788837459155231530996962, −7.55252614866431409822280885747, −6.34573398078978462188874853421, −5.78925088961090491963986425078, −4.05815419081526434768610302806, −2.72246932673929513528343412503,
1.19759543977822287810158099853, 3.64618217990703438800032190891, 4.39080535523133947534054514721, 5.30982133828597607699620673157, 6.97117499527756209803319290809, 8.293499772867525769464387121230, 9.684532708869799040348628147701, 10.34105295766984962084907758736, 11.19691265218238771286423772713, 12.21299251696815898731876184416
