L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−3.15 − 3.15i)7-s + 2.82·8-s − 2i·11-s + (1.60 + 1.60i)13-s + (6.09 − 1.63i)14-s + (−2.00 + 3.46i)16-s + (−4.24 + 4.24i)17-s + 3.19·19-s + (2.44 + 1.41i)22-s + (−5.27 + 5.27i)23-s + (−3.09 + 0.830i)26-s + (−2.31 + 8.62i)28-s + 0.535i·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.19 − 1.19i)7-s + 0.999·8-s − 0.603i·11-s + (0.444 + 0.444i)13-s + (1.62 − 0.436i)14-s + (−0.500 + 0.866i)16-s + (−1.02 + 1.02i)17-s + 0.733·19-s + (0.522 + 0.301i)22-s + (−1.10 + 1.10i)23-s + (−0.607 + 0.162i)26-s + (−0.436 + 1.62i)28-s + 0.0995i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0186190 + 0.284605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0186190 + 0.284605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.15 + 3.15i)T + 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-1.60 - 1.60i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + (5.27 - 5.27i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.535iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 + (7.72 - 7.72i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + (-3.15 + 3.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.656 + 0.656i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.86 - 3.86i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + (-3.91 - 3.91i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.53iT - 71T^{2} \) |
| 73 | \( 1 + (2.82 + 2.82i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + (-9.14 + 9.14i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (6.88 - 6.88i)T - 97iT^{2} \) |
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
−10.36261669043431176467975129446, −9.594962304898814583219808662971, −8.808692640517892845978561656518, −7.942614384285672359267005351785, −6.97810571064336677352133185060, −6.45928108594318765029417535467, −5.59337503203357923789573572345, −4.24265435529692897362871470726, −3.46782792682686906954721626908, −1.41496793317137727076854560065,
0.16406367318699070648338419252, 2.16257504369626554765742810610, 2.91052270960944335398644280555, 4.01760400436866315592602528340, 5.21170027207123456825774707621, 6.30738084973280265141821189772, 7.26352251173183958627297599255, 8.338095800222763537380201995302, 9.138671358747093169435214748114, 9.614572436634871937589088751120