| L(s) = 1 | − 27.2i·3-s − 279.·5-s − 3.32e3i·7-s + 5.81e3·9-s + 6.36e3i·11-s + 3.07e4·13-s + 7.61e3i·15-s + 1.22e5·17-s − 7.55e3i·19-s − 9.06e4·21-s + 4.06e5i·23-s + 7.81e4·25-s − 3.37e5i·27-s − 8.28e5·29-s + 1.39e6i·31-s + ⋯ |
| L(s) = 1 | − 0.336i·3-s − 0.447·5-s − 1.38i·7-s + 0.886·9-s + 0.434i·11-s + 1.07·13-s + 0.150i·15-s + 1.46·17-s − 0.0579i·19-s − 0.465·21-s + 1.45i·23-s + 0.200·25-s − 0.634i·27-s − 1.17·29-s + 1.51i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.426586157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426586157\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 279.T \) |
| good | 3 | \( 1 + 27.2iT - 6.56e3T^{2} \) |
| 7 | \( 1 + 3.32e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 6.36e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.07e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.22e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 7.55e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.06e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.39e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.86e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 9.72e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.61e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.87e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.01e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 6.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 9.79e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.19e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.62e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.35e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.22e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 6.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.58e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.97e7T + 7.83e15T^{2} \) |
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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
−10.28234397433221762617824194898, −9.479236875542400998390043775380, −8.024525202152438143795278471044, −7.42930023976520065324172284990, −6.68194077282461114467537117155, −5.29314534622042816949614217318, −4.02405731954334166483105136330, −3.44538132494109705016778077818, −1.48341339286142970202700776394, −0.957190284064567229060271828164,
0.62072710287105591199763040007, 1.93964199615977298050581333958, 3.23518421961919699164726233741, 4.16650334824713073751836255871, 5.43823645742812373528788072352, 6.18705326991688342092870482666, 7.53161531805598829047244080123, 8.469699376163418144385107918015, 9.233426944548369903123367837411, 10.24537935031140268628081234507
