| L(s) = 1 | + (0.230 + 1.30i)2-s + (−0.173 + 0.984i)3-s + (0.229 − 0.0835i)4-s + (1.21 + 1.02i)5-s − 1.32·6-s + (−2.25 − 1.89i)7-s + (1.48 + 2.57i)8-s + (−0.939 − 0.342i)9-s + (−1.05 + 1.82i)10-s + (−1.22 − 2.12i)11-s + (0.0424 + 0.240i)12-s + (−1.07 + 0.390i)13-s + (1.95 − 3.38i)14-s + (−1.21 + 1.02i)15-s + (−2.64 + 2.21i)16-s + (3.07 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.162 + 0.922i)2-s + (−0.100 + 0.568i)3-s + (0.114 − 0.0417i)4-s + (0.545 + 0.457i)5-s − 0.540·6-s + (−0.853 − 0.716i)7-s + (0.525 + 0.910i)8-s + (−0.313 − 0.114i)9-s + (−0.333 + 0.577i)10-s + (−0.369 − 0.640i)11-s + (0.0122 + 0.0694i)12-s + (−0.297 + 0.108i)13-s + (0.522 − 0.904i)14-s + (−0.314 + 0.264i)15-s + (−0.661 + 0.554i)16-s + (0.745 + 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0153 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0153 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.851931 + 0.838954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.851931 + 0.838954i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (-0.749 - 6.03i)T \) |
| good | 2 | \( 1 + (-0.230 - 1.30i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.21 - 1.02i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.25 + 1.89i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (1.22 + 2.12i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.07 - 0.390i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.07 - 1.11i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.00 + 5.68i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.230 + 0.398i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.100 + 0.174i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 41 | \( 1 + (9.28 - 3.37i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-3.00 + 5.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.57 - 8.03i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (10.8 - 9.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.52 + 3.46i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.33 - 6.99i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.536 + 3.04i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 - 3.91T + 73T^{2} \) |
| 79 | \( 1 + (8.04 + 6.75i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.08 - 2.21i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.11 - 4.29i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.01 + 6.95i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.97708428554683042618194741030, −13.38743603786893840526788926699, −11.66679788847767281026065953706, −10.52667545613446892971872205629, −9.859950490287165240436995055208, −8.307481627236452570817278164190, −6.94317742161936442715852422167, −6.17532761551499961352157328292, −4.90838812152935047281812903640, −3.02234751518694963992140132973,
1.85657270282156057740508994148, 3.25784088255866462059485974735, 5.30916798056468915356703228710, 6.58678634727580316376890573776, 7.88809792569775041467130951018, 9.520868996834863650286826789264, 10.16396063579536018975770320414, 11.63910287725825283186042201272, 12.49723643320652793968410641766, 12.86216352153605039306169295278
