L(s) = 1 | + (−2.03 − 1.17i)5-s + 4.52i·7-s − 2.20·11-s + (−1.93 − 3.35i)13-s + (−5.29 − 3.05i)17-s + (2.75 + 3.37i)19-s + (0.557 + 0.966i)23-s + (0.270 + 0.468i)25-s + (−3.51 + 2.02i)29-s − 6.92i·31-s + (5.33 − 9.23i)35-s + 10.9·37-s + (8.80 + 5.08i)41-s + (9.76 + 5.63i)43-s + (−6.43 − 11.1i)47-s + ⋯ |
L(s) = 1 | + (−0.911 − 0.526i)5-s + 1.71i·7-s − 0.666·11-s + (−0.536 − 0.929i)13-s + (−1.28 − 0.740i)17-s + (0.632 + 0.774i)19-s + (0.116 + 0.201i)23-s + (0.0541 + 0.0937i)25-s + (−0.652 + 0.376i)29-s − 1.24i·31-s + (0.900 − 1.56i)35-s + 1.80·37-s + (1.37 + 0.794i)41-s + (1.48 + 0.859i)43-s + (−0.938 − 1.62i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020692875\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020692875\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-2.75 - 3.37i)T \) |
good | 5 | \( 1 + (2.03 + 1.17i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 4.52iT - 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + (1.93 + 3.35i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (5.29 + 3.05i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.557 - 0.966i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.51 - 2.02i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + (-8.80 - 5.08i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.76 - 5.63i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.43 + 11.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.32 + 4.23i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.66 + 6.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.719 + 1.24i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.28 - 3.05i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.87 - 6.70i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.705 + 1.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.92 - 1.11i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + (-2.44 + 1.40i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.76 - 6.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.753430525178560078163055513461, −7.83773270855859122828498983601, −7.68740434730060488075881187210, −6.29664665365527676934825150285, −5.54832965785247625123111188531, −4.96811593539892753964026132038, −4.05112017199627097922643970051, −2.83902422904046196754045871633, −2.27101080237966397553240931568, −0.47765108936581251503856597559,
0.77370264303899438160827543515, 2.29291232853452149156999582588, 3.35505593453762038015552685958, 4.37680187185183754211058202367, 4.45616289328656137123441410873, 5.96782402276828745894864046976, 6.94263406508254744677723590328, 7.36343975601984436796787744786, 7.80793923566116514700437160620, 8.930705613246788733826628113954