| L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 2·13-s − 4·19-s − 4·21-s + 6·23-s + 4·27-s − 6·29-s + 4·31-s + 2·37-s + 4·39-s − 6·41-s + 10·43-s + 6·47-s − 3·49-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 − 0.554·13-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 + 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.225385985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225385985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.16087907793683, −14.70238880045124, −14.24657834649034, −13.53054146249441, −12.85051418263348, −12.54820926646322, −11.75987009959583, −11.53263167987284, −10.85080568098181, −10.63275038722811, −9.894986421571742, −9.211241408412817, −8.616399597807497, −8.058415744982228, −7.317945298954099, −6.827134258899572, −6.243141716851517, −5.395326750044494, −5.290555301495716, −4.456836757467861, −3.986484088738397, −2.879392482366463, −2.234973976227652, −1.284729040855391, −0.4951237445299724,
0.4951237445299724, 1.284729040855391, 2.234973976227652, 2.879392482366463, 3.986484088738397, 4.456836757467861, 5.290555301495716, 5.395326750044494, 6.243141716851517, 6.827134258899572, 7.317945298954099, 8.058415744982228, 8.616399597807497, 9.211241408412817, 9.894986421571742, 10.63275038722811, 10.85080568098181, 11.53263167987284, 11.75987009959583, 12.54820926646322, 12.85051418263348, 13.53054146249441, 14.24657834649034, 14.70238880045124, 15.16087907793683
