| 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
−10.29448739193929307287448705230, −9.496236669208801473395651898749, −9.064994706667829621776813565263, −8.310764084959290843648272482730, −7.05757770731097316446602719448, −6.16688659603535951240767595395, −4.86333488974442248393222023303, −3.85720322267679333217115494054, −2.67379416873222924066021641302, −0.66804739260402381524265705352,
1.71748942023707712969226847405, 2.80105142998164326797957661769, 3.66971356953516952991356117487, 5.93720871249641228150595152156, 6.67700157912854813029149060053, 7.08557575175052891685093135396, 8.347724514541018815858468447387, 9.224933718873033414552866141473, 9.720826873166418977213792859120, 10.78206926604445123377487554417
