| 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
−14.41285470432551836525586332149, −13.33579885810138715630312013463, −12.42354536932902358852139451655, −11.58427314502356676992672789308, −10.72578498659869196836034976545, −8.480574244429398683405592383587, −7.67981322465705254492978356642, −5.83506275083670387637702373807, −4.18266752064744035927116549833, −2.79570899128405437725245892643,
3.94555675701795956501059105764, 4.61629221355278721525381380927, 6.26145833122871900895372873577, 7.69615888582068597905934148510, 9.008866318083573694357099171090, 10.58031400985443845520648465701, 11.68723259903552499218171818506, 12.99541255205942942387417130338, 13.93599908496604562126489393715, 15.08658647555215136539314258095
