| L(s) = 1 | + 2·3-s + 2·7-s + 9-s + 6·17-s + 4·19-s + 4·21-s − 6·23-s − 4·27-s + 6·29-s + 4·31-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s − 12·59-s + 2·61-s + 2·63-s + 2·67-s − 12·69-s + 12·71-s + 2·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.25·23-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.244·67-s − 1.44·69-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.203847453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.203847453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.78812784542964, −15.25115479403371, −14.62218971780950, −14.17465762398010, −13.92325064422156, −13.39824380257110, −12.47075703141230, −12.03043126594466, −11.56227037346100, −10.78017453829248, −10.07256869843619, −9.666466810151499, −9.058468071259215, −8.272247941931688, −7.935122978007670, −7.640589517983138, −6.677792075393693, −5.920253280333684, −5.254253030363432, −4.578009367471638, −3.736117493923519, −3.196051486894461, −2.487838720123482, −1.707430277054586, −0.8536938349441409,
0.8536938349441409, 1.707430277054586, 2.487838720123482, 3.196051486894461, 3.736117493923519, 4.578009367471638, 5.254253030363432, 5.920253280333684, 6.677792075393693, 7.640589517983138, 7.935122978007670, 8.272247941931688, 9.058468071259215, 9.666466810151499, 10.07256869843619, 10.78017453829248, 11.56227037346100, 12.03043126594466, 12.47075703141230, 13.39824380257110, 13.92325064422156, 14.17465762398010, 14.62218971780950, 15.25115479403371, 15.78812784542964
