L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 11-s + 12-s − 4·14-s + 16-s − 8·17-s + 18-s + 19-s − 4·21-s − 22-s − 3·23-s + 24-s + 27-s − 4·28-s − 29-s + 31-s + 32-s − 33-s − 8·34-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.185·29-s + 0.179·31-s + 0.176·32-s − 0.174·33-s − 1.37·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.531417878744045818807268828279, −7.45654365124277994811016836720, −6.70756534175442960648465476997, −6.30080590719207160691412332497, −5.24306934453883269904147412330, −4.30801399964828431306303840795, −3.53698332852679765889221274218, −2.80683694786922496217827209499, −1.93092679700969988135790454085, 0,
1.93092679700969988135790454085, 2.80683694786922496217827209499, 3.53698332852679765889221274218, 4.30801399964828431306303840795, 5.24306934453883269904147412330, 6.30080590719207160691412332497, 6.70756534175442960648465476997, 7.45654365124277994811016836720, 8.531417878744045818807268828279