L(s) = 1 | + (−1.41 − 0.0912i)2-s + (1.98 + 0.257i)4-s + (−1.86 − 1.86i)7-s + (−2.77 − 0.544i)8-s + 0.728i·11-s + (3.12 + 3.12i)13-s + (2.46 + 2.80i)14-s + (3.86 + 1.02i)16-s + (1.12 − 1.12i)17-s − 3.73·19-s + (0.0664 − 1.02i)22-s + (5.83 − 5.83i)23-s + (−4.12 − 4.69i)26-s + (−3.22 − 4.18i)28-s − 2.64i·29-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0645i)2-s + (0.991 + 0.128i)4-s + (−0.705 − 0.705i)7-s + (−0.981 − 0.192i)8-s + 0.219i·11-s + (0.866 + 0.866i)13-s + (0.658 + 0.749i)14-s + (0.966 + 0.255i)16-s + (0.272 − 0.272i)17-s − 0.856·19-s + (0.0141 − 0.219i)22-s + (1.21 − 1.21i)23-s + (−0.808 − 0.920i)26-s + (−0.608 − 0.790i)28-s − 0.490i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.703227 - 0.453910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.703227 - 0.453910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0912i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.86 + 1.86i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.728iT - 11T^{2} \) |
| 13 | \( 1 + (-3.12 - 3.12i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + (-5.83 + 5.83i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.64iT - 29T^{2} \) |
| 31 | \( 1 + 6.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.12 - 3.12i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.09 + 2.09i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.484 - 0.484i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + (5.10 + 5.10i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 + (3.96 + 3.96i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 + (3.55 - 3.55i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.03iT - 89T^{2} \) |
| 97 | \( 1 + (-12.5 + 12.5i)T - 97iT^{2} \) |
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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
−9.932962485606847017005942132105, −9.112102299229268275321108171598, −8.468251010247376283245822600049, −7.39110571292555900210491846927, −6.70096255410907261900330292323, −6.03277822807771933417174262347, −4.42781402500668444575919039729, −3.37953445478973567693104237047, −2.12703923853798285635001810457, −0.62808533172042881038902463637,
1.20386637964534183572350461898, 2.71401246015654760405497885370, 3.54542466380634388470658615827, 5.38697223191440664297720986417, 6.05930412540661707276960888982, 6.93009252609939468523904297690, 7.895138805937115921516119175413, 8.775658428750587678118947235429, 9.212645576030818897444521736133, 10.25514417872047484393555635086