L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.28 − 1.16i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.362 − 1.69i)6-s − 0.535·7-s − 0.999·8-s + (0.304 + 2.98i)9-s + (−0.866 − 0.499i)10-s − 5.20i·11-s + (1.64 − 0.532i)12-s + (1.58 + 0.917i)13-s + (−0.267 − 0.463i)14-s + (1.69 + 0.362i)15-s + (−0.5 − 0.866i)16-s + (3.93 − 2.27i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.742 − 0.670i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.148 − 0.691i)6-s − 0.202·7-s − 0.353·8-s + (0.101 + 0.994i)9-s + (−0.273 − 0.158i)10-s − 1.57i·11-s + (0.475 − 0.153i)12-s + (0.440 + 0.254i)13-s + (−0.0715 − 0.123i)14-s + (0.437 + 0.0936i)15-s + (−0.125 − 0.216i)16-s + (0.954 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.824288 - 0.484772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.824288 - 0.484772i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.28 + 1.16i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (1.25 + 4.17i)T \) |
good | 7 | \( 1 + 0.535T + 7T^{2} \) |
| 11 | \( 1 + 5.20iT - 11T^{2} \) |
| 13 | \( 1 + (-1.58 - 0.917i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.93 + 2.27i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (5.55 + 3.20i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.17iT - 31T^{2} \) |
| 37 | \( 1 - 5.32iT - 37T^{2} \) |
| 41 | \( 1 + (2.33 + 4.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.21 - 7.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.52 + 2.03i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.26 + 5.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.47 - 6.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.11 + 7.12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.445 + 0.257i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.29 + 5.71i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.05 - 5.29i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.65 + 4.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 + (8.15 - 14.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.567 + 0.327i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.88557771911947429220446587206, −9.735769594253905149644947691933, −8.327677537340297985378979297646, −7.947597114722438476729560821538, −6.67657283281122246271733755598, −6.20367803517028074692225293367, −5.26029089210254524309148963750, −4.04959974678060943861419485083, −2.74765699087325832994743224215, −0.56385745455046746776519190078,
1.53442584217101003443071635241, 3.42939441647792348834344499267, 4.19383548418521534087971949982, 5.16574914319665321834262332880, 6.02893428700863192057902514651, 7.18687831644193111037759081377, 8.393824422088026399307600727234, 9.548700024320804471607834729970, 10.19070640794632631774022653751, 10.75190181305146936515750847999