| L(s) = 1 | + (1.34 + 0.861i)2-s + (0.142 + 0.989i)3-s + (0.225 + 0.492i)4-s + (−2.43 − 0.715i)5-s + (−0.662 + 1.45i)6-s + (0.729 − 0.841i)7-s + (0.330 − 2.30i)8-s + (−0.959 + 0.281i)9-s + (−2.65 − 3.05i)10-s + (−1.19 + 0.768i)11-s + (−0.455 + 0.293i)12-s + (2.76 + 3.19i)13-s + (1.70 − 0.500i)14-s + (0.361 − 2.51i)15-s + (3.13 − 3.62i)16-s + (−2.04 + 4.47i)17-s + ⋯ |
| L(s) = 1 | + (0.948 + 0.609i)2-s + (0.0821 + 0.571i)3-s + (0.112 + 0.246i)4-s + (−1.08 − 0.319i)5-s + (−0.270 + 0.592i)6-s + (0.275 − 0.317i)7-s + (0.116 − 0.813i)8-s + (−0.319 + 0.0939i)9-s + (−0.838 − 0.967i)10-s + (−0.360 + 0.231i)11-s + (−0.131 + 0.0845i)12-s + (0.768 + 0.886i)13-s + (0.455 − 0.133i)14-s + (0.0932 − 0.648i)15-s + (0.784 − 0.905i)16-s + (−0.495 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13814 + 0.475342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13814 + 0.475342i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.44 - 4.12i)T \) |
| good | 2 | \( 1 + (-1.34 - 0.861i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (2.43 + 0.715i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.729 + 0.841i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (1.19 - 0.768i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 3.19i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.04 - 4.47i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.40 + 5.26i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.66 + 5.84i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.05 - 7.36i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.94 + 2.33i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-4.41 - 1.29i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.288 + 2.00i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + (-6.82 + 7.88i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (7.84 + 9.05i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.58 - 10.9i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (9.58 + 6.15i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.26 - 2.09i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.12 + 11.2i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (3.56 + 4.11i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.50 + 0.440i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.324 - 2.26i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-8.39 - 2.46i)T + (81.6 + 52.4i)T^{2} \) |
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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
−15.08658647555215136539314258095, −13.93599908496604562126489393715, −12.99541255205942942387417130338, −11.68723259903552499218171818506, −10.58031400985443845520648465701, −9.008866318083573694357099171090, −7.69615888582068597905934148510, −6.26145833122871900895372873577, −4.61629221355278721525381380927, −3.94555675701795956501059105764,
2.79570899128405437725245892643, 4.18266752064744035927116549833, 5.83506275083670387637702373807, 7.67981322465705254492978356642, 8.480574244429398683405592383587, 10.72578498659869196836034976545, 11.58427314502356676992672789308, 12.42354536932902358852139451655, 13.33579885810138715630312013463, 14.41285470432551836525586332149
