| L(s) = 1 | − 21.9·3-s + 235.·7-s + 238.·9-s − 561.·11-s − 118.·13-s + 1.53e3·17-s + 1.95e3·19-s − 5.16e3·21-s − 494.·23-s + 94.8·27-s + 3.86e3·29-s + 9.55e3·31-s + 1.23e4·33-s − 4.80e3·37-s + 2.59e3·39-s − 1.54e4·41-s − 1.02e4·43-s + 4.26e3·47-s + 3.84e4·49-s − 3.36e4·51-s − 3.39e3·53-s − 4.28e4·57-s − 2.79e4·59-s + 1.52e4·61-s + 5.61e4·63-s − 2.29e4·67-s + 1.08e4·69-s + ⋯ |
| L(s) = 1 | − 1.40·3-s + 1.81·7-s + 0.982·9-s − 1.39·11-s − 0.194·13-s + 1.28·17-s + 1.24·19-s − 2.55·21-s − 0.194·23-s + 0.0250·27-s + 0.854·29-s + 1.78·31-s + 1.96·33-s − 0.576·37-s + 0.273·39-s − 1.43·41-s − 0.842·43-s + 0.281·47-s + 2.28·49-s − 1.80·51-s − 0.165·53-s − 1.74·57-s − 1.04·59-s + 0.523·61-s + 1.78·63-s − 0.623·67-s + 0.274·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.585849837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585849837\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 21.9T + 243T^{2} \) |
| 7 | \( 1 - 235.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 561.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 118.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.53e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 494.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.64e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.83e4T + 8.58e9T^{2} \) |
| 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
−9.940696279643754664651711679727, −8.292873096425845558244410369914, −7.897077529816216552345471637713, −6.90454847691309908115594389074, −5.63670847514596262536018311661, −5.16574745371459122927764242318, −4.62295413042602477604508845858, −2.95118985289249571586161977187, −1.53387858943092683431540502392, −0.66707040694906220946816781176,
0.66707040694906220946816781176, 1.53387858943092683431540502392, 2.95118985289249571586161977187, 4.62295413042602477604508845858, 5.16574745371459122927764242318, 5.63670847514596262536018311661, 6.90454847691309908115594389074, 7.897077529816216552345471637713, 8.292873096425845558244410369914, 9.940696279643754664651711679727
