L(s) = 1 | − 2·4-s + 5-s + 7-s − 3·11-s + 4·16-s + 3·17-s + 4·19-s − 2·20-s + 9·23-s + 25-s − 2·28-s + 6·29-s − 2·31-s + 35-s + 37-s − 3·41-s + 2·43-s + 6·44-s − 6·47-s − 6·49-s − 9·53-s − 3·55-s − 12·59-s + 5·61-s − 8·64-s + 4·67-s − 6·68-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s − 0.904·11-s + 16-s + 0.727·17-s + 0.917·19-s − 0.447·20-s + 1.87·23-s + 1/5·25-s − 0.377·28-s + 1.11·29-s − 0.359·31-s + 0.169·35-s + 0.164·37-s − 0.468·41-s + 0.304·43-s + 0.904·44-s − 0.875·47-s − 6/7·49-s − 1.23·53-s − 0.404·55-s − 1.56·59-s + 0.640·61-s − 64-s + 0.488·67-s − 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.801221470\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.801221470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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
−7.899852757808751196863738266821, −7.37847609144377513735920143106, −6.41094706066728876391037150030, −5.56895029981771839658716727502, −4.96922748610343118432089441920, −4.66190946366239145964821100773, −3.34184374186132703964531735348, −2.95068554311414819768064013502, −1.59745923283821507583674432057, −0.71817739384042992252886876843,
0.71817739384042992252886876843, 1.59745923283821507583674432057, 2.95068554311414819768064013502, 3.34184374186132703964531735348, 4.66190946366239145964821100773, 4.96922748610343118432089441920, 5.56895029981771839658716727502, 6.41094706066728876391037150030, 7.37847609144377513735920143106, 7.899852757808751196863738266821