| L(s) = 1 | − 2-s + 0.561·3-s + 4-s − 3.29·5-s − 0.561·6-s − 3.86·7-s − 8-s − 2.68·9-s + 3.29·10-s + 0.120·11-s + 0.561·12-s − 1.45·13-s + 3.86·14-s − 1.85·15-s + 16-s − 5.48·17-s + 2.68·18-s − 7.85·19-s − 3.29·20-s − 2.16·21-s − 0.120·22-s + 2.95·23-s − 0.561·24-s + 5.87·25-s + 1.45·26-s − 3.18·27-s − 3.86·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.323·3-s + 0.5·4-s − 1.47·5-s − 0.229·6-s − 1.46·7-s − 0.353·8-s − 0.895·9-s + 1.04·10-s + 0.0364·11-s + 0.161·12-s − 0.402·13-s + 1.03·14-s − 0.477·15-s + 0.250·16-s − 1.32·17-s + 0.632·18-s − 1.80·19-s − 0.737·20-s − 0.472·21-s − 0.0257·22-s + 0.615·23-s − 0.114·24-s + 1.17·25-s + 0.284·26-s − 0.613·27-s − 0.730·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08323839006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08323839006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.561T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 - 0.120T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 8.11T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
| 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
−8.997532344005257240643147999821, −8.110330124949378660916089877703, −7.49340604903949686813562928152, −6.59228647834728393995995016518, −6.18726863839763316108995619116, −4.73130519223061004369378959776, −3.81650449608781110369208299511, −3.10461156491978516620114049260, −2.24753276221796688856173315148, −0.17881887448031119907827787053,
0.17881887448031119907827787053, 2.24753276221796688856173315148, 3.10461156491978516620114049260, 3.81650449608781110369208299511, 4.73130519223061004369378959776, 6.18726863839763316108995619116, 6.59228647834728393995995016518, 7.49340604903949686813562928152, 8.110330124949378660916089877703, 8.997532344005257240643147999821
