L(s) = 1 | − 3-s − 7-s + 9-s − 3·11-s − 2·13-s + 4·17-s − 2·19-s + 21-s + 9·23-s − 27-s − 7·29-s + 10·31-s + 3·33-s − 5·37-s + 2·39-s − 8·41-s + 7·43-s + 10·47-s + 49-s − 4·51-s − 10·53-s + 2·57-s − 6·59-s − 8·61-s − 63-s + 13·67-s − 9·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.970·17-s − 0.458·19-s + 0.218·21-s + 1.87·23-s − 0.192·27-s − 1.29·29-s + 1.79·31-s + 0.522·33-s − 0.821·37-s + 0.320·39-s − 1.24·41-s + 1.06·43-s + 1.45·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.264·57-s − 0.781·59-s − 1.02·61-s − 0.125·63-s + 1.58·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−7.899473909663191807572903112462, −7.27720131383344427771370987871, −6.61351801028347888938198222606, −5.70297484775777887400292195384, −5.16017802130727569371052854335, −4.40891702746943177258309030421, −3.28568334008219714322200718461, −2.57096142035794862492835757112, −1.24547113208558814577415151922, 0,
1.24547113208558814577415151922, 2.57096142035794862492835757112, 3.28568334008219714322200718461, 4.40891702746943177258309030421, 5.16017802130727569371052854335, 5.70297484775777887400292195384, 6.61351801028347888938198222606, 7.27720131383344427771370987871, 7.899473909663191807572903112462