L(s) = 1 | − 1.76·5-s + (1.85 + 1.88i)7-s + 6.12·11-s + (−0.380 + 0.658i)13-s + (3.42 − 5.92i)17-s + (−0.971 − 1.68i)19-s − 0.421·23-s − 1.89·25-s + (−0.732 − 1.26i)29-s + (3.85 + 6.67i)31-s + (−3.26 − 3.32i)35-s + (1.44 + 2.49i)37-s + (3.47 − 6.01i)41-s + (−4.33 − 7.49i)43-s + (−0.830 + 1.43i)47-s + ⋯ |
L(s) = 1 | − 0.787·5-s + (0.699 + 0.714i)7-s + 1.84·11-s + (−0.105 + 0.182i)13-s + (0.829 − 1.43i)17-s + (−0.222 − 0.385i)19-s − 0.0877·23-s − 0.379·25-s + (−0.135 − 0.235i)29-s + (0.691 + 1.19i)31-s + (−0.551 − 0.562i)35-s + (0.237 + 0.410i)37-s + (0.542 − 0.939i)41-s + (−0.660 − 1.14i)43-s + (−0.121 + 0.209i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.964625082\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.964625082\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.85 - 1.88i)T \) |
good | 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.33 + 7.49i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.39 - 5.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.56 + 2.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.81 + 3.14i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.714297036974460374303484465752, −8.054630136299510439554257990924, −7.17522018303721267968357245676, −6.62445251261970066351018317831, −5.58149192285459349391795291836, −4.80243154287378268862938324661, −4.02099081526398689432690925668, −3.18626104204633165493883524305, −2.01492531660253585232327081576, −0.862182705804347412208479469874,
0.954338697660650758644828365425, 1.81208602696050159770034749459, 3.42875879472191157855926364381, 4.03128301947096485354020855155, 4.51664047719054178576162137638, 5.83016712084265005356912335173, 6.44041711485531466409360826961, 7.36565794902752576639873492405, 8.023732401312006071232191294585, 8.466409850803858274359039688595