| L(s) = 1 | + (3.67 + 2.12i)5-s + (−2 − 3.46i)7-s + (−14.6 + 8.48i)11-s + (−4 + 6.92i)13-s − 12.7i·17-s + 16·19-s + (−14.6 − 8.48i)23-s + (−3.5 − 6.06i)25-s + (3.67 − 2.12i)29-s + (22 − 38.1i)31-s − 16.9i·35-s − 34·37-s + (−40.4 − 23.3i)41-s + (−20 − 34.6i)43-s + (73.4 − 42.4i)47-s + ⋯ |
| L(s) = 1 | + (0.734 + 0.424i)5-s + (−0.285 − 0.494i)7-s + (−1.33 + 0.771i)11-s + (−0.307 + 0.532i)13-s − 0.748i·17-s + 0.842·19-s + (−0.638 − 0.368i)23-s + (−0.140 − 0.242i)25-s + (0.126 − 0.0731i)29-s + (0.709 − 1.22i)31-s − 0.484i·35-s − 0.918·37-s + (−0.985 − 0.569i)41-s + (−0.465 − 0.805i)43-s + (1.56 − 0.902i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9752676337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9752676337\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.67 - 2.12i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (2 + 3.46i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (14.6 - 8.48i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4 - 6.92i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + (14.6 + 8.48i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-3.67 + 2.12i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-22 + 38.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 + (40.4 + 23.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (20 + 34.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-73.4 + 42.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-29.3 - 16.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25 + 43.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 16T + 5.32e3T^{2} \) |
| 79 | \( 1 + (38 + 65.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (102. - 59.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (88 + 152. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.454936790424449515763914422079, −8.379630847958156366856789518543, −7.35565364849390748525416387409, −6.94596637347013372918513786552, −5.82077498142723927320191538175, −5.06154193222875592479225219724, −4.04649999288718766986586578256, −2.75254598387340981625669053704, −2.03142903507950591933560985407, −0.27191087641061814251547865218,
1.31350147956933695268378227765, 2.59348341771777709382306834740, 3.39676378052576679278060865521, 4.91356047486478775495262055357, 5.55465288284498307877700565553, 6.13214046530207413774425331884, 7.36460429992047752345623876811, 8.224866825720574609331927805546, 8.829208730818436988612464815232, 9.847701865130698899617972610478
