L(s) = 1 | + (−1.40 + 0.136i)2-s + (−1.50 + 0.855i)3-s + (1.96 − 0.382i)4-s − 2.98·5-s + (2.00 − 1.40i)6-s + 1.36i·7-s + (−2.71 + 0.806i)8-s + (1.53 − 2.57i)9-s + (4.19 − 0.405i)10-s − 5.80i·11-s + (−2.62 + 2.25i)12-s − 0.852i·13-s + (−0.185 − 1.91i)14-s + (4.48 − 2.54i)15-s + (3.70 − 1.50i)16-s + 5.38i·17-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0961i)2-s + (−0.869 + 0.493i)3-s + (0.981 − 0.191i)4-s − 1.33·5-s + (0.818 − 0.575i)6-s + 0.514i·7-s + (−0.958 + 0.285i)8-s + (0.512 − 0.858i)9-s + (1.32 − 0.128i)10-s − 1.74i·11-s + (−0.759 + 0.651i)12-s − 0.236i·13-s + (−0.0495 − 0.512i)14-s + (1.15 − 0.657i)15-s + (0.926 − 0.375i)16-s + 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.408581 + 0.174026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.408581 + 0.174026i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.136i)T \) |
| 3 | \( 1 + (1.50 - 0.855i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 1.36iT - 7T^{2} \) |
| 11 | \( 1 + 5.80iT - 11T^{2} \) |
| 13 | \( 1 + 0.852iT - 13T^{2} \) |
| 17 | \( 1 - 5.38iT - 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 + 1.32iT - 31T^{2} \) |
| 37 | \( 1 - 0.210iT - 37T^{2} \) |
| 41 | \( 1 - 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 4.46iT - 59T^{2} \) |
| 61 | \( 1 - 14.3iT - 61T^{2} \) |
| 67 | \( 1 + 9.77T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + 4.99iT - 79T^{2} \) |
| 83 | \( 1 - 3.09iT - 83T^{2} \) |
| 89 | \( 1 + 13.7iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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
−10.80724201032096632196292827637, −10.30053052605623888869316747705, −8.877768332711918514264322953689, −8.454921554028299257931998591864, −7.48040743472796219261947999482, −6.26429596282245106981757079883, −5.69900488704226130574231734097, −4.14652287174155414002248814638, −3.09115339345234298003178582274, −0.804909518418301423507020489972,
0.61818670872946137127297849860, 2.24186478973680754342588419456, 4.00284255014651730743039366947, 5.02682223271533139072897186082, 6.63972629754740965845020791454, 7.32571544887352930968676871353, 7.61718824069699617975586434814, 8.875563552462191512281669354613, 9.979354381097275928193756815872, 10.69477350700683008128199557760