L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 11-s + 12-s + 13-s + 14-s + 16-s − 3·17-s + 18-s + 21-s + 22-s − 3·23-s + 24-s + 26-s + 27-s + 28-s + 5·29-s − 4·31-s + 32-s + 33-s − 3·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.218·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.928·29-s − 0.718·31-s + 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84734002345151, −12.23880420428459, −11.91982263114201, −11.35239116639130, −11.03836289769916, −10.46466754128037, −10.12406884342961, −9.371488358559697, −9.190375203099864, −8.495853508070450, −8.169411408551769, −7.608879259821442, −7.274077637030303, −6.605142815788875, −6.095806401662463, −5.974681368193559, −4.922851281988930, −4.772822249761146, −4.249062152062154, −3.703623081496403, −3.171924831543000, −2.725719233801802, −2.005610401722817, −1.661989415196432, −0.9575678824331757, 0,
0.9575678824331757, 1.661989415196432, 2.005610401722817, 2.725719233801802, 3.171924831543000, 3.703623081496403, 4.249062152062154, 4.772822249761146, 4.922851281988930, 5.974681368193559, 6.095806401662463, 6.605142815788875, 7.274077637030303, 7.608879259821442, 8.169411408551769, 8.495853508070450, 9.190375203099864, 9.371488358559697, 10.12406884342961, 10.46466754128037, 11.03836289769916, 11.35239116639130, 11.91982263114201, 12.23880420428459, 12.84734002345151