| 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
−12.14947372554925653851980623592, −11.19015393413831594107187930178, −10.28486019950748108280448776610, −8.505628132299661063278817370274, −8.051000744064590943808263277139, −7.09373204068613572343923113008, −5.24005218927700893125136804207, −4.95707539703682876426468950943, −1.71341474561782469128192481857, −0.12863332146796702010775349552,
3.65184410729454148587361052473, 4.72943054411122650293939155613, 5.57111353298861253306574253310, 7.38411308901697602428324772991, 8.352419650329903322267316789624, 9.739911118268096447763924382509, 10.42866961230512896981239155487, 11.06542507940083816543821945177, 11.85897784248344327662465338451, 13.42274050681258480635120303784
