| L(s) = 1 | − 3.25·3-s + 7.62·9-s + 11-s − 5.29·13-s − 2.03·17-s − 4.48·19-s + 7.62·23-s − 15.0·27-s − 5.62·29-s − 1.22·31-s − 3.25·33-s − 7.62·37-s + 17.2·39-s + 9.77·41-s − 11.6·43-s − 6.51·47-s + 6.62·51-s + 11.2·53-s + 14.6·57-s − 5.29·59-s + 7.74·61-s − 8.24·67-s − 24.8·69-s + 2.37·71-s + 16.2·73-s + 5.62·79-s + 26.2·81-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 2.54·9-s + 0.301·11-s − 1.46·13-s − 0.492·17-s − 1.02·19-s + 1.58·23-s − 2.90·27-s − 1.04·29-s − 0.220·31-s − 0.567·33-s − 1.25·37-s + 2.76·39-s + 1.52·41-s − 1.77·43-s − 0.950·47-s + 0.927·51-s + 1.54·53-s + 1.93·57-s − 0.688·59-s + 0.991·61-s − 1.00·67-s − 2.99·69-s + 0.282·71-s + 1.90·73-s + 0.632·79-s + 2.91·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4636280042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4636280042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 7.62T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 7.62T + 37T^{2} \) |
| 41 | \( 1 - 9.77T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 0.804T + 83T^{2} \) |
| 89 | \( 1 + 0.804T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
| 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
−7.22457674084419348343307917685, −6.92838790575280312520336142558, −6.38157050137617669919175449022, −5.46316053778971824177840189587, −5.08268978469961938004537153574, −4.47848552412884312035981604202, −3.69337322645432867222923269437, −2.39563822009889538797079567213, −1.49475659547766379652651811782, −0.36656218033564945048951429922,
0.36656218033564945048951429922, 1.49475659547766379652651811782, 2.39563822009889538797079567213, 3.69337322645432867222923269437, 4.47848552412884312035981604202, 5.08268978469961938004537153574, 5.46316053778971824177840189587, 6.38157050137617669919175449022, 6.92838790575280312520336142558, 7.22457674084419348343307917685
