L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s + 9-s + 2·12-s − 2·13-s − 2·14-s − 4·16-s + 4·17-s − 2·18-s + 3·19-s + 21-s − 2·23-s + 4·26-s + 27-s + 2·28-s − 6·29-s − 5·31-s + 8·32-s − 8·34-s + 2·36-s − 3·37-s − 6·38-s − 2·39-s + 2·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 16-s + 0.970·17-s − 0.471·18-s + 0.688·19-s + 0.218·21-s − 0.417·23-s + 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.898·31-s + 1.41·32-s − 1.37·34-s + 1/3·36-s − 0.493·37-s − 0.973·38-s − 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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
−7.59531957988214722888502169056, −7.25791325281328535801450824438, −6.20354659970605176182323361609, −5.37102234217169302590584245258, −4.57411096858554217724319789573, −3.68637954248596662751659906930, −2.80563283065460247959383296377, −1.88480143611337159351659625120, −1.24023099237540172023227223112, 0,
1.24023099237540172023227223112, 1.88480143611337159351659625120, 2.80563283065460247959383296377, 3.68637954248596662751659906930, 4.57411096858554217724319789573, 5.37102234217169302590584245258, 6.20354659970605176182323361609, 7.25791325281328535801450824438, 7.59531957988214722888502169056