| L(s) = 1 | + (−0.802 + 0.0420i)2-s + (−1.97 + 2.43i)3-s + (−1.34 + 0.141i)4-s + (−1.30 + 1.81i)5-s + (1.48 − 2.03i)6-s + (2.52 − 0.782i)7-s + (2.66 − 0.421i)8-s + (−1.42 − 6.69i)9-s + (0.974 − 1.50i)10-s + (−4.90 − 1.04i)11-s + (2.31 − 3.56i)12-s + (−0.816 + 0.416i)13-s + (−1.99 + 0.734i)14-s + (−1.83 − 6.77i)15-s + (0.531 − 0.112i)16-s + (0.185 − 0.483i)17-s + ⋯ |
| L(s) = 1 | + (−0.567 + 0.0297i)2-s + (−1.14 + 1.40i)3-s + (−0.673 + 0.0707i)4-s + (−0.585 + 0.810i)5-s + (0.605 − 0.832i)6-s + (0.955 − 0.295i)7-s + (0.941 − 0.149i)8-s + (−0.474 − 2.23i)9-s + (0.308 − 0.477i)10-s + (−1.47 − 0.314i)11-s + (0.668 − 1.02i)12-s + (−0.226 + 0.115i)13-s + (−0.533 + 0.196i)14-s + (−0.474 − 1.74i)15-s + (0.132 − 0.0282i)16-s + (0.0450 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0480367 - 0.0859202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0480367 - 0.0859202i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.30 - 1.81i)T \) |
| 7 | \( 1 + (-2.52 + 0.782i)T \) |
| good | 2 | \( 1 + (0.802 - 0.0420i)T + (1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (1.97 - 2.43i)T + (-0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (4.90 + 1.04i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.816 - 0.416i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.185 + 0.483i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.116 - 1.10i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.270 - 5.16i)T + (-22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (3.39 + 4.67i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.139 + 0.312i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (5.35 + 3.47i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (6.75 + 2.19i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (3.02 - 3.02i)T - 43iT^{2} \) |
| 47 | \( 1 + (11.9 - 4.57i)T + (34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-4.55 - 3.68i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-5.88 - 6.53i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (1.07 + 0.969i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 0.504i)T + (49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (6.16 - 4.47i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.656 + 1.01i)T + (-29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.97 + 4.44i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.543 - 3.42i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (2.10 - 2.33i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.557 + 3.51i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.42274050681258480635120303784, −11.85897784248344327662465338451, −11.06542507940083816543821945177, −10.42866961230512896981239155487, −9.739911118268096447763924382509, −8.352419650329903322267316789624, −7.38411308901697602428324772991, −5.57111353298861253306574253310, −4.72943054411122650293939155613, −3.65184410729454148587361052473,
0.12863332146796702010775349552, 1.71341474561782469128192481857, 4.95707539703682876426468950943, 5.24005218927700893125136804207, 7.09373204068613572343923113008, 8.051000744064590943808263277139, 8.505628132299661063278817370274, 10.28486019950748108280448776610, 11.19015393413831594107187930178, 12.14947372554925653851980623592
