L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·11-s − 13-s + 14-s + 16-s − 4·19-s − 3·22-s − 3·23-s − 26-s + 28-s + 6·29-s − 4·31-s + 32-s − 37-s − 4·38-s + 6·41-s + 8·43-s − 3·44-s − 3·46-s − 3·47-s + 49-s − 52-s − 12·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.639·22-s − 0.625·23-s − 0.196·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.164·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.452·44-s − 0.442·46-s − 0.437·47-s + 1/7·49-s − 0.138·52-s − 1.64·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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
−7.41330077363837754322091166494, −6.52133364998610111979308043800, −5.94265269717602928614809617413, −5.23351504820406829963381335104, −4.55980854323130659082988841742, −4.01147967442140803093248015635, −2.95310480436136814549878386314, −2.38532196269207462194180119688, −1.44695512510906848770433468118, 0,
1.44695512510906848770433468118, 2.38532196269207462194180119688, 2.95310480436136814549878386314, 4.01147967442140803093248015635, 4.55980854323130659082988841742, 5.23351504820406829963381335104, 5.94265269717602928614809617413, 6.52133364998610111979308043800, 7.41330077363837754322091166494