L(s) = 1 | − 3·2-s − 3·3-s + 4-s + 9·6-s + 7·7-s + 21·8-s + 9·9-s − 6·11-s − 3·12-s + 41·13-s − 21·14-s − 71·16-s + 27·17-s − 27·18-s − 4·19-s − 21·21-s + 18·22-s + 75·23-s − 63·24-s − 123·26-s − 27·27-s + 7·28-s − 123·29-s − 205·31-s + 45·32-s + 18·33-s − 81·34-s + ⋯ |
L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.612·6-s + 0.377·7-s + 0.928·8-s + 1/3·9-s − 0.164·11-s − 0.0721·12-s + 0.874·13-s − 0.400·14-s − 1.10·16-s + 0.385·17-s − 0.353·18-s − 0.0482·19-s − 0.218·21-s + 0.174·22-s + 0.679·23-s − 0.535·24-s − 0.927·26-s − 0.192·27-s + 0.0472·28-s − 0.787·29-s − 1.18·31-s + 0.248·32-s + 0.0949·33-s − 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8008020404\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8008020404\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 41 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 75 T + p^{3} T^{2} \) |
| 29 | \( 1 + 123 T + p^{3} T^{2} \) |
| 31 | \( 1 + 205 T + p^{3} T^{2} \) |
| 37 | \( 1 + 262 T + p^{3} T^{2} \) |
| 41 | \( 1 - 57 T + p^{3} T^{2} \) |
| 43 | \( 1 - 407 T + p^{3} T^{2} \) |
| 47 | \( 1 + 60 T + p^{3} T^{2} \) |
| 53 | \( 1 - 327 T + p^{3} T^{2} \) |
| 59 | \( 1 - 33 T + p^{3} T^{2} \) |
| 61 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 628 T + p^{3} T^{2} \) |
| 71 | \( 1 - 300 T + p^{3} T^{2} \) |
| 73 | \( 1 - 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 686 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1401 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 494 T + p^{3} T^{2} \) |
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
−10.67063996959682623289740934649, −9.426828544454660550860376835728, −8.825674353503716207137782928288, −7.82044232836892699189332152629, −7.10422665453445385765281558672, −5.84618544103517681974772894550, −4.88851012646406030824006012141, −3.70566697569315546484787937350, −1.81312822846185337724341518398, −0.69118954983381603792449980743,
0.69118954983381603792449980743, 1.81312822846185337724341518398, 3.70566697569315546484787937350, 4.88851012646406030824006012141, 5.84618544103517681974772894550, 7.10422665453445385765281558672, 7.82044232836892699189332152629, 8.825674353503716207137782928288, 9.426828544454660550860376835728, 10.67063996959682623289740934649