L(s) = 1 | + (0.933 + 2.66i)2-s + (−3.69 − 3.69i)3-s + (−6.25 + 4.98i)4-s + (6.41 − 13.3i)6-s + (19.3 − 19.3i)7-s + (−19.1 − 12.0i)8-s + 0.287i·9-s + 23.4i·11-s + (41.5 + 4.69i)12-s + (62.9 − 62.9i)13-s + (69.7 + 33.5i)14-s + (14.2 − 62.3i)16-s + (−0.872 − 0.872i)17-s + (−0.768 + 0.268i)18-s + 57.0·19-s + ⋯ |
L(s) = 1 | + (0.330 + 0.943i)2-s + (−0.710 − 0.710i)3-s + (−0.782 + 0.623i)4-s + (0.436 − 0.905i)6-s + (1.04 − 1.04i)7-s + (−0.846 − 0.532i)8-s + 0.0106i·9-s + 0.641i·11-s + (0.998 + 0.112i)12-s + (1.34 − 1.34i)13-s + (1.33 + 0.641i)14-s + (0.223 − 0.974i)16-s + (−0.0124 − 0.0124i)17-s + (−0.0100 + 0.00351i)18-s + 0.688·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.34044 - 0.303350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34044 - 0.303350i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.933 - 2.66i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (3.69 + 3.69i)T + 27iT^{2} \) |
| 7 | \( 1 + (-19.3 + 19.3i)T - 343iT^{2} \) |
| 11 | \( 1 - 23.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-62.9 + 62.9i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (0.872 + 0.872i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 57.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + (116. + 116. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 121. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 66.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (36.6 + 36.6i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 302.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-190. - 190. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-54.1 + 54.1i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (36.5 - 36.5i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 401.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-187. + 187. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 584. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (436. - 436. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 608.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-124. - 124. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 684. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-384. - 384. i)T + 9.12e5iT^{2} \) |
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
−13.38844551755893135659291540941, −12.50010756944029632892857956793, −11.40811699880104017423427223714, −10.16929437193553671922576321971, −8.358159214868836511093425016900, −7.54732072106689674100662714584, −6.46683092776233697297284181855, −5.34102277285598741678290958494, −3.93648308840266552925542615739, −0.858314055116101834916484838929,
1.77865971169596005255741123997, 3.82835872700854510836060039776, 5.11641453275015856797530918455, 5.93209540783476147373549615191, 8.376576018920409164831799487704, 9.312597403263430584906068812723, 10.62214798950823794158678791532, 11.57051223058542776165233434822, 11.75717669287684417470383698773, 13.50750678008892460567060103681