| L(s) = 1 | + 4.52i·2-s − 12.4·4-s + 1.26·5-s − 20.1i·8-s + 5.72i·10-s + 41.5i·11-s − 85.7i·13-s − 8.54·16-s − 77.7·17-s − 48.6i·19-s − 15.7·20-s − 188.·22-s − 90.9i·23-s − 123.·25-s + 387.·26-s + ⋯ |
| L(s) = 1 | + 1.59i·2-s − 1.55·4-s + 0.113·5-s − 0.890i·8-s + 0.181i·10-s + 1.14i·11-s − 1.82i·13-s − 0.133·16-s − 1.10·17-s − 0.587i·19-s − 0.176·20-s − 1.82·22-s − 0.824i·23-s − 0.987·25-s + 2.92·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6940195625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6940195625\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.52iT - 8T^{2} \) |
| 5 | \( 1 - 1.26T + 125T^{2} \) |
| 11 | \( 1 - 41.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 85.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 77.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 90.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 151. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 88.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 90.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 334. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 880.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 13.1iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 341. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 921. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 413.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 954.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 29.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40002404558050345694812665578, −9.548779681802306336987868865804, −8.505325863123758027551298924924, −7.78828505056965787051273809846, −6.96064851662718378635505696515, −6.07513434470893357156774673071, −5.14116579801432021074887123920, −4.25781845736438651132815593849, −2.45185810030302730183816737198, −0.22714681955871365266122683747,
1.37216262772317272089426712356, 2.38291126327462303392227475828, 3.66770081988597398990728377672, 4.41452747333063952823495085657, 5.86555798106109389133344446336, 6.99936086385106542415447129706, 8.487642458611108238865198570015, 9.226617680372255975285333718650, 9.914770134704048166767184060921, 11.17171600618549752030080881939
