L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.917 − 1.46i)3-s + (0.939 + 0.342i)4-s + (0.501 + 2.17i)5-s + (−1.15 + 1.28i)6-s + (−3.34 + 1.93i)7-s + (−0.866 − 0.5i)8-s + (−1.31 − 2.69i)9-s + (−0.115 − 2.23i)10-s + (3.47 + 2.00i)11-s + (1.36 − 1.06i)12-s + (−0.681 + 0.571i)13-s + (3.63 − 1.32i)14-s + (3.66 + 1.26i)15-s + (0.766 + 0.642i)16-s + (−0.566 + 3.21i)17-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.529 − 0.848i)3-s + (0.469 + 0.171i)4-s + (0.224 + 0.974i)5-s + (−0.473 + 0.525i)6-s + (−1.26 + 0.730i)7-s + (−0.306 − 0.176i)8-s + (−0.438 − 0.898i)9-s + (−0.0366 − 0.706i)10-s + (1.04 + 0.605i)11-s + (0.394 − 0.307i)12-s + (−0.189 + 0.158i)13-s + (0.970 − 0.353i)14-s + (0.945 + 0.326i)15-s + (0.191 + 0.160i)16-s + (−0.137 + 0.778i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.874483 + 0.483436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.874483 + 0.483436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-0.917 + 1.46i)T \) |
| 5 | \( 1 + (-0.501 - 2.17i)T \) |
| 19 | \( 1 + (3.42 - 2.69i)T \) |
good | 7 | \( 1 + (3.34 - 1.93i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.47 - 2.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.681 - 0.571i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.566 - 3.21i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.82 - 2.48i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 6.16i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.50 + 0.868i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.85T + 37T^{2} \) |
| 41 | \( 1 + (4.43 + 3.72i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.819 - 2.25i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 8.51i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.88 + 10.6i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.00803 - 0.0455i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.60 + 0.946i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.646 - 3.66i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 4.35i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.68 - 9.15i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.79 + 9.29i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.50 - 2.60i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.50 - 5.45i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 13.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.78206926604445123377487554417, −9.720826873166418977213792859120, −9.224933718873033414552866141473, −8.347724514541018815858468447387, −7.08557575175052891685093135396, −6.67700157912854813029149060053, −5.93720871249641228150595152156, −3.66971356953516952991356117487, −2.80105142998164326797957661769, −1.71748942023707712969226847405,
0.66804739260402381524265705352, 2.67379416873222924066021641302, 3.85720322267679333217115494054, 4.86333488974442248393222023303, 6.16688659603535951240767595395, 7.05757770731097316446602719448, 8.310764084959290843648272482730, 9.064994706667829621776813565263, 9.496236669208801473395651898749, 10.29448739193929307287448705230