| L(s) = 1 | + 6.27·3-s − 10.7·5-s − 6.71·7-s + 12.3·9-s + 58.1·11-s − 0.679·13-s − 67.6·15-s − 6.24·17-s + 87.8·19-s − 42.1·21-s − 8.74·25-s − 92.0·27-s + 85.9·29-s − 196.·31-s + 364.·33-s + 72.4·35-s + 191.·37-s − 4.25·39-s − 78.3·41-s + 16.6·43-s − 132.·45-s + 244.·47-s − 297.·49-s − 39.1·51-s − 345.·53-s − 626.·55-s + 550.·57-s + ⋯ |
| L(s) = 1 | + 1.20·3-s − 0.964·5-s − 0.362·7-s + 0.456·9-s + 1.59·11-s − 0.0144·13-s − 1.16·15-s − 0.0890·17-s + 1.06·19-s − 0.437·21-s − 0.0699·25-s − 0.656·27-s + 0.550·29-s − 1.13·31-s + 1.92·33-s + 0.349·35-s + 0.848·37-s − 0.0174·39-s − 0.298·41-s + 0.0591·43-s − 0.440·45-s + 0.758·47-s − 0.868·49-s − 0.107·51-s − 0.895·53-s − 1.53·55-s + 1.27·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.923554524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923554524\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 6.27T + 27T^{2} \) |
| 5 | \( 1 + 10.7T + 125T^{2} \) |
| 7 | \( 1 + 6.71T + 343T^{2} \) |
| 11 | \( 1 - 58.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.679T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.24T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 85.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 78.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 16.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 244.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 345.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 763.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 636.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 559.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 839.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 718.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 935.T + 9.12e5T^{2} \) |
| 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
−8.699753577059892151840450857864, −8.067432122516436651033578178701, −7.31979280518454959752441708344, −6.65879942344897667944741192818, −5.58298450798127551513205535469, −4.30448372386105716138753632580, −3.68165623189551211571207407367, −3.11392986347385062175122356578, −1.93437612723498496814750335193, −0.73839434920766174642160112159,
0.73839434920766174642160112159, 1.93437612723498496814750335193, 3.11392986347385062175122356578, 3.68165623189551211571207407367, 4.30448372386105716138753632580, 5.58298450798127551513205535469, 6.65879942344897667944741192818, 7.31979280518454959752441708344, 8.067432122516436651033578178701, 8.699753577059892151840450857864
