| L(s) = 1 | + (604. + 349. i)5-s + (1.12e3 + 1.94e3i)7-s + (1.88e4 − 1.08e4i)11-s + (−9.20e3 + 1.59e4i)13-s + 5.65e4i·17-s + 2.12e5·19-s + (1.43e4 + 8.31e3i)23-s + (4.83e4 + 8.37e4i)25-s + (2.94e5 − 1.70e5i)29-s + (8.29e4 − 1.43e5i)31-s + 1.56e6i·35-s + 1.11e6·37-s + (3.60e6 + 2.08e6i)41-s + (−1.28e6 − 2.23e6i)43-s + (8.10e6 − 4.67e6i)47-s + ⋯ |
| L(s) = 1 | + (0.967 + 0.558i)5-s + (0.468 + 0.811i)7-s + (1.28 − 0.742i)11-s + (−0.322 + 0.558i)13-s + 0.676i·17-s + 1.62·19-s + (0.0514 + 0.0297i)23-s + (0.123 + 0.214i)25-s + (0.417 − 0.240i)29-s + (0.0897 − 0.155i)31-s + 1.04i·35-s + 0.596·37-s + (1.27 + 0.736i)41-s + (−0.376 − 0.652i)43-s + (1.66 − 0.958i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.883109140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.883109140\) |
| \(L(5)\) |
|
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 | \( 1 + (-604. - 349. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-1.12e3 - 1.94e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.88e4 + 1.08e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (9.20e3 - 1.59e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 5.65e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.12e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.43e4 - 8.31e3i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-2.94e5 + 1.70e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-8.29e4 + 1.43e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.11e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.60e6 - 2.08e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (1.28e6 + 2.23e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-8.10e6 + 4.67e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 4.75e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (8.39e6 + 4.84e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.04e6 + 5.27e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.02e7 + 1.77e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.08e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 9.02e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + (1.67e7 + 2.89e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (5.09e7 - 2.94e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 8.65e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.68e7 + 8.12e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−9.718767111311078259435219437095, −9.201111680384227850405519325025, −8.228278782798605975667958265688, −7.00227821201594779050104277561, −6.09569310176068946871493217454, −5.48836030042679571239096104184, −4.13574708535730455488419886275, −2.92478959113878390764232243274, −1.96082798507514182463033583016, −0.998064115587919149257848025776,
0.901631843057730455808114106364, 1.38318873287760934233687515111, 2.71460399530755450069753906694, 4.08970628355486924251583761763, 4.96748079533850593757584518278, 5.86363245230586015110223411721, 7.06446150298813029121935714751, 7.71944964881741287488475982677, 9.145160567247269764729004752711, 9.551137079853386185997183656431
