L(s) = 1 | + (9.93 + 9.93i)2-s + (27.3 + 27.3i)3-s − 58.4i·4-s − 447. i·5-s + 543. i·6-s + 2.07e3·7-s + (3.12e3 − 3.12e3i)8-s − 5.06e3i·9-s + (4.44e3 − 4.44e3i)10-s + (5.20e3 + 5.20e3i)11-s + (1.59e3 − 1.59e3i)12-s + 1.39e4i·13-s + (2.06e4 + 2.06e4i)14-s + (1.22e4 − 1.22e4i)15-s + 4.71e4·16-s + (621. + 621. i)17-s + ⋯ |
L(s) = 1 | + (0.621 + 0.621i)2-s + (0.337 + 0.337i)3-s − 0.228i·4-s − 0.716i·5-s + 0.419i·6-s + 0.865·7-s + (0.762 − 0.762i)8-s − 0.772i·9-s + (0.444 − 0.444i)10-s + (0.355 + 0.355i)11-s + (0.0770 − 0.0770i)12-s + 0.488i·13-s + (0.537 + 0.537i)14-s + (0.241 − 0.241i)15-s + 0.719·16-s + (0.00743 + 0.00743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00769i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 + 0.00769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(3.02945 - 0.0116502i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02945 - 0.0116502i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (6.93e5 + 1.39e5i)T \) |
good | 2 | \( 1 + (-9.93 - 9.93i)T + 256iT^{2} \) |
| 3 | \( 1 + (-27.3 - 27.3i)T + 6.56e3iT^{2} \) |
| 5 | \( 1 + 447. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 2.07e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (-5.20e3 - 5.20e3i)T + 2.14e8iT^{2} \) |
| 13 | \( 1 - 1.39e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-621. - 621. i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + (-2.14e4 - 2.14e4i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 - 1.10e5T + 7.83e10T^{2} \) |
| 31 | \( 1 + (674. + 674. i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (1.23e6 - 1.23e6i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + (1.64e6 - 1.64e6i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (2.44e6 + 2.44e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (1.34e6 - 1.34e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 - 1.63e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.86e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + (-1.12e7 - 1.12e7i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 - 1.90e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.15e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (2.37e7 - 2.37e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + (2.06e7 + 2.06e7i)T + 1.51e15iT^{2} \) |
| 83 | \( 1 + 1.71e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-5.27e7 - 5.27e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (4.78e7 - 4.78e7i)T - 7.83e15iT^{2} \) |
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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
−15.01589007261598026268655927765, −14.42918556292398506526073901164, −13.09835699353231241423213239006, −11.67201399440074548538042576884, −9.913656277484119205806101467198, −8.647435389463089193082684042284, −6.88707127081740579654628023071, −5.24250008880720799352349791135, −4.09034050395085376587134279640, −1.29412578874645745140455984435,
1.90638405684661504494731694968, 3.28676340927968478713391553031, 5.05139960376905995233420919757, 7.30046449475968927695313381134, 8.426629299795457512855272820776, 10.67801475987948470244262773562, 11.47495942449384888023352551044, 12.90892702590620106854148425641, 13.92684563122493607716125061152, 14.78855424339834200316896495080