| L(s) = 1 | + (0.212 − 0.977i)2-s + (1.62 − 1.21i)3-s + (−0.909 − 0.415i)4-s + (0.606 + 2.15i)5-s + (−0.840 − 1.84i)6-s + (0.178 − 2.48i)7-s + (−0.599 + 0.800i)8-s + (0.309 − 1.05i)9-s + (2.23 − 0.134i)10-s + (2.81 − 4.37i)11-s + (−1.97 + 0.430i)12-s + (−1.36 + 0.0979i)13-s + (−2.39 − 0.703i)14-s + (3.59 + 2.75i)15-s + (0.654 + 0.755i)16-s + (−1.90 + 5.11i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (0.935 − 0.700i)3-s + (−0.454 − 0.207i)4-s + (0.271 + 0.962i)5-s + (−0.343 − 0.751i)6-s + (0.0672 − 0.940i)7-s + (−0.211 + 0.283i)8-s + (0.103 − 0.351i)9-s + (0.705 − 0.0426i)10-s + (0.848 − 1.32i)11-s + (−0.571 + 0.124i)12-s + (−0.379 + 0.0271i)13-s + (−0.639 − 0.187i)14-s + (0.927 + 0.710i)15-s + (0.163 + 0.188i)16-s + (−0.462 + 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31542 - 1.09462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31542 - 1.09462i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (-0.606 - 2.15i)T \) |
| 23 | \( 1 + (3.49 - 3.28i)T \) |
| good | 3 | \( 1 + (-1.62 + 1.21i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (-0.178 + 2.48i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 4.37i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.36 - 0.0979i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (1.90 - 5.11i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-0.418 + 0.916i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.914 - 0.417i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.224 - 1.56i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.63 - 6.66i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-5.36 + 1.57i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (6.52 + 8.71i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (1.51 - 1.51i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.04 - 0.289i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-4.89 - 4.23i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (8.75 - 1.25i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (15.3 + 3.34i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-4.18 + 2.69i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (1.41 - 0.528i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (7.48 - 8.63i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-10.4 + 5.71i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-2.03 + 14.1i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-5.90 - 3.22i)T + (52.4 + 81.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93792364054843532814277287805, −10.97573990585367047078276216215, −10.27852575965761991212878630807, −9.053218475950291231827010535118, −8.079591833873815728396819022464, −7.05391536001535487848037440509, −5.97845514835154469344511277377, −3.96428438066982509169681704150, −3.04764196921210695049649276622, −1.65430681092011208726412382596,
2.38651264398381558122430788186, 4.16720708397760513696301749650, 4.91851552202284946055442024272, 6.22393137963087334520774959972, 7.62321242523348339518508875406, 8.703407867774925422145176567734, 9.360895620790681647531592883390, 9.809594157980611403948556737161, 11.83176327167529338514552427134, 12.45325976176189284243334193645
