L(s) = 1 | + (−0.611 − 1.27i)2-s + 3.21·3-s + (−1.25 + 1.56i)4-s − 5-s + (−1.96 − 4.09i)6-s − 3.02i·7-s + (2.75 + 0.640i)8-s + 7.30·9-s + (0.611 + 1.27i)10-s + 0.488i·11-s + (−4.01 + 5.00i)12-s − 0.943i·13-s + (−3.86 + 1.85i)14-s − 3.21·15-s + (−0.869 − 3.90i)16-s + 3.16·17-s + ⋯ |
L(s) = 1 | + (−0.432 − 0.901i)2-s + 1.85·3-s + (−0.625 + 0.780i)4-s − 0.447·5-s + (−0.802 − 1.67i)6-s − 1.14i·7-s + (0.974 + 0.226i)8-s + 2.43·9-s + (0.193 + 0.403i)10-s + 0.147i·11-s + (−1.15 + 1.44i)12-s − 0.261i·13-s + (−1.03 + 0.495i)14-s − 0.828·15-s + (−0.217 − 0.976i)16-s + 0.768·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41999 - 1.09230i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41999 - 1.09230i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.611 + 1.27i)T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (2.58 - 3.50i)T \) |
good | 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 + 3.02iT - 7T^{2} \) |
| 11 | \( 1 - 0.488iT - 11T^{2} \) |
| 13 | \( 1 + 0.943iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 23 | \( 1 + 7.08iT - 23T^{2} \) |
| 29 | \( 1 - 8.80iT - 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 1.11iT - 37T^{2} \) |
| 41 | \( 1 - 1.56iT - 41T^{2} \) |
| 43 | \( 1 + 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 0.471iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 + 8.62T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 18.0iT - 83T^{2} \) |
| 89 | \( 1 - 9.83iT - 89T^{2} \) |
| 97 | \( 1 + 1.71iT - 97T^{2} \) |
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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
−10.66200679096062333390269093910, −10.33800751736307266383536383379, −9.269310319651400086374195089184, −8.475026989386514992815821384775, −7.76320465187196873407829877262, −7.08637456754047455411034268625, −4.49812434150299475445477433320, −3.73738102176294996420888300445, −2.87887695221089379499452324330, −1.42980157425830407484887403142,
1.96024940294666458348385314496, 3.34772591273115948436903329119, 4.57155332774930705572362434006, 5.95770378339800031962096809649, 7.29834798973310221190247764721, 7.913955492014899714676826454352, 8.757214930133166686197193274063, 9.264960789719907461468880244556, 10.05602522460208049542027270372, 11.52407526346795428898657338241