| L(s) = 1 | + (−1.86 − 0.724i)2-s + (2.95 + 2.69i)4-s + (−0.693 − 1.20i)5-s + (−0.562 + 0.975i)7-s + (−3.54 − 7.17i)8-s + (0.422 + 2.73i)10-s + (1.11 − 1.92i)11-s + (14.4 − 8.34i)13-s + (1.75 − 1.41i)14-s + (1.42 + 15.9i)16-s + 20.2i·17-s − 21.0i·19-s + (1.19 − 5.41i)20-s + (−3.46 + 2.78i)22-s + (18.0 − 10.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.932 − 0.362i)2-s + (0.737 + 0.674i)4-s + (−0.138 − 0.240i)5-s + (−0.0804 + 0.139i)7-s + (−0.443 − 0.896i)8-s + (0.0422 + 0.273i)10-s + (0.101 − 0.175i)11-s + (1.11 − 0.642i)13-s + (0.125 − 0.100i)14-s + (0.0889 + 0.996i)16-s + 1.19i·17-s − 1.10i·19-s + (0.0597 − 0.270i)20-s + (−0.157 + 0.126i)22-s + (0.784 − 0.452i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.860503 - 0.486970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860503 - 0.486970i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.86 + 0.724i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.693 + 1.20i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.562 - 0.975i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-1.11 + 1.92i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-14.4 + 8.34i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 20.2iT - 289T^{2} \) |
| 19 | \( 1 + 21.0iT - 361T^{2} \) |
| 23 | \( 1 + (-18.0 + 10.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-26.9 + 46.7i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-9.19 - 15.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-15.1 + 8.72i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (44.2 + 25.5i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.4 + 18.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-27.1 - 46.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-76.5 - 44.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (66.0 - 38.1i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 69.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.8 + 34.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (56.9 - 98.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 25.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-10.4 + 18.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.75118313617207982400991007813, −10.81677496994538366368101653129, −10.10473433097570589988177454379, −8.685713368993606837989922310300, −8.421081858279634396385244002322, −6.98646836592768079504257389912, −5.93965323308573332867672144310, −4.10288823965056019059150936701, −2.69199144895584449707355368712, −0.871736387329460033210381931447,
1.35414888823447939937816787724, 3.22902935174658580782589951445, 5.07817809510966039162813685879, 6.43682301816575591262475552550, 7.19441003685133058124348386721, 8.352482206127553906739638267348, 9.233476850031124801496932335943, 10.18720362768285052989863208676, 11.17880365433321281109703315561, 11.85745168962236463681207949978
