L(s) = 1 | + 2.60·2-s + 3-s + 4.80·4-s − 0.565·5-s + 2.60·6-s + 7.32·8-s + 9-s − 1.47·10-s + 2.31·11-s + 4.80·12-s + 13-s − 0.565·15-s + 9.50·16-s + 3.03·17-s + 2.60·18-s − 8.50·19-s − 2.71·20-s + 6.05·22-s + 2.33·23-s + 7.32·24-s − 4.68·25-s + 2.60·26-s + 27-s + 4.06·29-s − 1.47·30-s + 0.245·31-s + 10.1·32-s + ⋯ |
L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.40·4-s − 0.252·5-s + 1.06·6-s + 2.59·8-s + 0.333·9-s − 0.466·10-s + 0.699·11-s + 1.38·12-s + 0.277·13-s − 0.145·15-s + 2.37·16-s + 0.736·17-s + 0.615·18-s − 1.95·19-s − 0.607·20-s + 1.29·22-s + 0.486·23-s + 1.49·24-s − 0.936·25-s + 0.511·26-s + 0.192·27-s + 0.754·29-s − 0.269·30-s + 0.0440·31-s + 1.79·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.703902977\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.703902977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 0.565T + 5T^{2} \) |
| 11 | \( 1 - 2.31T + 11T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 + 8.50T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 0.245T + 31T^{2} \) |
| 37 | \( 1 - 0.180T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 7.37T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 5.06T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 6.88T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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
−9.104088103668510612671389274646, −8.237202150406974352509952518409, −7.38941505457303461363272563667, −6.53691964218130710588487357923, −6.01633386678946640669252150548, −4.92276192356313005487559334605, −4.15474987903849962362905157752, −3.58404412583352911081190564222, −2.63586365205553203957023559549, −1.63283213968166080994241378839,
1.63283213968166080994241378839, 2.63586365205553203957023559549, 3.58404412583352911081190564222, 4.15474987903849962362905157752, 4.92276192356313005487559334605, 6.01633386678946640669252150548, 6.53691964218130710588487357923, 7.38941505457303461363272563667, 8.237202150406974352509952518409, 9.104088103668510612671389274646