L(s) = 1 | + (1.5 − 0.866i)5-s + 2.64·7-s + (3.96 + 2.29i)11-s + 3.46i·13-s + (−4.5 − 2.59i)17-s + (1.32 + 2.29i)19-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s + (−1.32 + 2.29i)31-s + (3.96 − 2.29i)35-s + (−3.5 − 6.06i)37-s + 3.46i·41-s − 9.16i·43-s + (3.96 + 6.87i)47-s + 7.00·49-s + ⋯ |
L(s) = 1 | + (0.670 − 0.387i)5-s + 0.999·7-s + (1.19 + 0.690i)11-s + 0.960i·13-s + (−1.09 − 0.630i)17-s + (0.303 + 0.525i)19-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s + (−0.237 + 0.411i)31-s + (0.670 − 0.387i)35-s + (−0.575 − 0.996i)37-s + 0.541i·41-s − 1.39i·43-s + (0.578 + 1.00i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.115665273\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.115665273\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.96 - 2.29i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.96 - 2.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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
−9.783168150100075503798626046598, −9.100523809772954772429508176414, −8.615683952140882336754887959745, −7.27556897760004803016544550522, −6.74439535691993880593196688754, −5.56603854341519971068724878277, −4.71846521011306307977979935820, −3.95333522225174300825059342518, −2.20974371425420272304315663410, −1.40949386715488600927214932143,
1.20590323516215075444832225952, 2.42433849975433527769903920059, 3.64106911829651362794749256939, 4.74605152955831451290131254109, 5.72377639599791445657553349901, 6.47662414254567900730104175892, 7.40219822223365340816047759884, 8.459174659670076817924752917333, 8.989646640230076948410052316194, 10.00966489573113557064137425659