L(s) = 1 | + (1.77 + 1.02i)3-s + (1.48 − 2.18i)7-s + (0.594 + 1.02i)9-s + (1.78 − 3.08i)11-s − 2.90i·13-s + (−3.22 − 1.85i)17-s + (−1.78 − 3.09i)19-s + (4.87 − 2.35i)21-s + (2.69 − 1.55i)23-s − 3.70i·27-s − 1.57·29-s + (0.382 − 0.661i)31-s + (6.31 − 3.64i)33-s + (−6.06 + 3.50i)37-s + (2.97 − 5.15i)39-s + ⋯ |
L(s) = 1 | + (1.02 + 0.590i)3-s + (0.562 − 0.826i)7-s + (0.198 + 0.343i)9-s + (0.537 − 0.930i)11-s − 0.806i·13-s + (−0.781 − 0.450i)17-s + (−0.410 − 0.711i)19-s + (1.06 − 0.513i)21-s + (0.561 − 0.324i)23-s − 0.713i·27-s − 0.293·29-s + (0.0686 − 0.118i)31-s + (1.09 − 0.634i)33-s + (−0.996 + 0.575i)37-s + (0.476 − 0.825i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.406225691\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.406225691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.48 + 2.18i)T \) |
good | 3 | \( 1 + (-1.77 - 1.02i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.78 + 3.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.90iT - 13T^{2} \) |
| 17 | \( 1 + (3.22 + 1.85i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 + 3.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.69 + 1.55i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 + (-0.382 + 0.661i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 - 3.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (6.38 - 3.68i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.88 - 1.66i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.07 + 7.04i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.96 - 5.14i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.1 - 7.00i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + (-11.4 - 6.61i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.49 - 2.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + (-2.35 - 4.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.397310306185607794806621881698, −8.589772076390466830890447303198, −8.168419917102242001030919493097, −7.14400247942372596368413757279, −6.27941758550730456988641522888, −5.02882059406305489050494511315, −4.21761495404611784924150669147, −3.38583547885533441895787762150, −2.51895552028586012559589134824, −0.874487854571375923713135398797,
1.87204863846944343618142485202, 2.07640852437930450358095208743, 3.50238343792918314183437038580, 4.48381607190971464863451288596, 5.50095583686221325180261413377, 6.62582100598417554785681712728, 7.29271647319487481775955212058, 8.144992179222038393280768145470, 8.899835026921767568386948051607, 9.219246254876525354630577003049