L(s) = 1 | + (−0.932 + 1.61i)3-s + 3.93·7-s + (−0.240 − 0.416i)9-s − 2.01·11-s + (−3.09 − 5.36i)13-s + (−2.28 + 3.95i)17-s + (−4.29 + 0.721i)19-s + (−3.66 + 6.34i)21-s + (−4.32 − 7.48i)23-s − 4.70·27-s + (2.59 + 4.49i)29-s − 0.856·31-s + (1.88 − 3.25i)33-s − 6.03·37-s + 11.5·39-s + ⋯ |
L(s) = 1 | + (−0.538 + 0.932i)3-s + 1.48·7-s + (−0.0800 − 0.138i)9-s − 0.608·11-s + (−0.859 − 1.48i)13-s + (−0.553 + 0.958i)17-s + (−0.986 + 0.165i)19-s + (−0.800 + 1.38i)21-s + (−0.901 − 1.56i)23-s − 0.904·27-s + (0.482 + 0.835i)29-s − 0.153·31-s + (0.327 − 0.567i)33-s − 0.991·37-s + 1.85·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3111071028\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3111071028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.29 - 0.721i)T \) |
good | 3 | \( 1 + (0.932 - 1.61i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + (3.09 + 5.36i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.32 + 7.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 - 4.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.856T + 31T^{2} \) |
| 37 | \( 1 + 6.03T + 37T^{2} \) |
| 41 | \( 1 + (1.37 - 2.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.39 - 11.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.32 - 7.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.17 + 5.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.18 - 12.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.73 - 8.20i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.24 + 9.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.51 - 4.35i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.663 + 1.14i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.38 + 2.39i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + (3.45 + 5.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.02 + 3.50i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.02968562333351663057834423138, −8.668389384303005836680575766495, −8.186913632573896985117008725953, −7.52625754342022694359978561498, −6.24535657322142405367742964361, −5.43138377245842626572374495035, −4.68463429904179072069751259955, −4.32612242526575884619130641549, −2.85563755281691213441415062341, −1.74565516648966075403719322804,
0.11206887802578637142619545873, 1.76699253950554310844072609479, 2.15606230017571546897625907907, 3.95360616791706063958444003789, 4.84159855172067269312920368475, 5.47969180891959675278032625532, 6.59967329496591600218687651721, 7.18507803066343278999740065688, 7.80180623438634767422645744601, 8.656897024575163381416399201606