L(s) = 1 | + 3-s + 2·7-s + 9-s − 2·11-s − 2·13-s + 2·17-s + 2·19-s + 2·21-s + 2·23-s + 27-s + 6·29-s − 4·31-s − 2·33-s + 2·37-s − 2·39-s − 10·41-s + 8·43-s + 2·47-s − 3·49-s + 2·51-s + 6·53-s + 2·57-s + 2·59-s + 10·61-s + 2·63-s − 8·67-s + 2·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 0.417·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s + 1.21·43-s + 0.291·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.264·57-s + 0.260·59-s + 1.28·61-s + 0.251·63-s − 0.977·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.870650969\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.870650969\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−7.61295942439814775303227056642, −7.28678416462206897733045665610, −6.40611248629867975050791270467, −5.41311726225445212290122181860, −5.00072082159812255691516870223, −4.21242687629678847339117748795, −3.32775706472927611987098554627, −2.61312897531506630225860802666, −1.82823716075282964321793118883, −0.797167509131121261696928555606,
0.797167509131121261696928555606, 1.82823716075282964321793118883, 2.61312897531506630225860802666, 3.32775706472927611987098554627, 4.21242687629678847339117748795, 5.00072082159812255691516870223, 5.41311726225445212290122181860, 6.40611248629867975050791270467, 7.28678416462206897733045665610, 7.61295942439814775303227056642