L(s) = 1 | + (1.69 − 0.368i)3-s + (1.80 + 1.93i)7-s + (2.72 − 1.24i)9-s + (4.05 − 2.34i)11-s − 2.18i·13-s + (−3.74 − 6.49i)17-s + (−0.638 − 0.368i)19-s + (3.76 + 2.61i)21-s + (−6.99 − 4.03i)23-s + (4.15 − 3.11i)27-s + 1.15i·29-s + (8.95 − 5.16i)31-s + (6.00 − 5.45i)33-s + (−2.30 + 3.99i)37-s + (−0.806 − 3.70i)39-s + ⋯ |
L(s) = 1 | + (0.977 − 0.212i)3-s + (0.680 + 0.732i)7-s + (0.909 − 0.415i)9-s + (1.22 − 0.706i)11-s − 0.607i·13-s + (−0.909 − 1.57i)17-s + (−0.146 − 0.0845i)19-s + (0.820 + 0.571i)21-s + (−1.45 − 0.842i)23-s + (0.800 − 0.599i)27-s + 0.214i·29-s + (1.60 − 0.928i)31-s + (1.04 − 0.950i)33-s + (−0.379 + 0.657i)37-s + (−0.129 − 0.593i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.891335323\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.891335323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 + 0.368i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.80 - 1.93i)T \) |
good | 11 | \( 1 + (-4.05 + 2.34i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.18iT - 13T^{2} \) |
| 17 | \( 1 + (3.74 + 6.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.638 + 0.368i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.99 + 4.03i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.15iT - 29T^{2} \) |
| 31 | \( 1 + (-8.95 + 5.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.30 - 3.99i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + (4.34 - 7.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.03 - 4.06i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.48 - 6.04i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.13 - 2.96i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.691 + 1.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.26iT - 71T^{2} \) |
| 73 | \( 1 + (-0.211 + 0.122i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.79 - 10.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + (-0.658 + 1.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.84iT - 97T^{2} \) |
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
−8.992275056303339614858817318706, −8.260875742302279000113087931787, −7.72998534114604305713177080834, −6.62583103209321244964455559973, −6.05977729966754551119553226307, −4.78314424468324339267229514444, −4.11163992409668255799579683117, −2.91908631508725951248014180530, −2.26876869534461527453598169755, −0.974079906933383124825489692419,
1.54802180175432593176238664564, 2.07106842349228417585034679236, 3.75001280643326740981960945193, 4.05676356237461454309272456847, 4.84370923352964708261900838953, 6.31353848949291925069335165774, 6.90095477178913691890100421514, 7.81551932952986070625600303499, 8.417545787690125354821025360943, 9.101383091736948998706086333935