L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 3·11-s + 12-s + 2·13-s + 14-s + 15-s + 16-s − 18-s + 5·19-s + 20-s − 21-s + 3·22-s + 9·23-s − 24-s + 25-s − 2·26-s + 27-s − 28-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.639·22-s + 1.87·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.497817869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.497817869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−9.739304715770159510083314826393, −9.399703805966851743005930503040, −8.445092458894771426974887958607, −7.67898912598372569763165919472, −6.87173655406222785546940528966, −5.87317845623954297947973895113, −4.85347326439029846408470834907, −3.31523095659482692669656609861, −2.57508659135638847507806336328, −1.11390817959686257456313576170,
1.11390817959686257456313576170, 2.57508659135638847507806336328, 3.31523095659482692669656609861, 4.85347326439029846408470834907, 5.87317845623954297947973895113, 6.87173655406222785546940528966, 7.67898912598372569763165919472, 8.445092458894771426974887958607, 9.399703805966851743005930503040, 9.739304715770159510083314826393