| L(s) = 1 | + 3·3-s + 5·5-s + 0.942·7-s + 9·9-s + 64.4·11-s − 77.8·13-s + 15·15-s + 73.5·17-s + 72.8·19-s + 2.82·21-s + 23·23-s + 25·25-s + 27·27-s + 18.1·29-s − 95.4·31-s + 193.·33-s + 4.71·35-s − 244.·37-s − 233.·39-s + 102.·41-s + 213.·43-s + 45·45-s + 465.·47-s − 342.·49-s + 220.·51-s + 227.·53-s + 322.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.0509·7-s + 0.333·9-s + 1.76·11-s − 1.66·13-s + 0.258·15-s + 1.04·17-s + 0.880·19-s + 0.0293·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.116·29-s − 0.552·31-s + 1.01·33-s + 0.0227·35-s − 1.08·37-s − 0.958·39-s + 0.390·41-s + 0.756·43-s + 0.149·45-s + 1.44·47-s − 0.997·49-s + 0.606·51-s + 0.590·53-s + 0.789·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.429331526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.429331526\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 0.942T + 343T^{2} \) |
| 11 | \( 1 - 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 18.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 465.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 227.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 345.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 152.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 200.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 787.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 167.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 824.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
| 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
−9.427753375053525022124019664954, −8.536208648156767508236865040484, −7.41094652080579208210496722122, −7.02315277950453007651021767667, −5.87721127620504024209058873954, −5.00120650759501131797947030583, −3.97196373631844067184727619014, −3.06001404612540635716563122129, −1.97155925410565298074148301062, −0.943889194901126807846837584300,
0.943889194901126807846837584300, 1.97155925410565298074148301062, 3.06001404612540635716563122129, 3.97196373631844067184727619014, 5.00120650759501131797947030583, 5.87721127620504024209058873954, 7.02315277950453007651021767667, 7.41094652080579208210496722122, 8.536208648156767508236865040484, 9.427753375053525022124019664954
