| L(s) = 1 | + (23.7 + 77.4i)3-s + (705. − 407. i)5-s + (−2.16e3 − 1.04e3i)7-s + (−5.43e3 + 3.67e3i)9-s + (1.90e4 + 1.10e4i)11-s − 5.33e4·13-s + (4.82e4 + 4.49e4i)15-s + (7.89e4 + 4.56e4i)17-s + (8.22e4 + 1.42e5i)19-s + (2.91e4 − 1.92e5i)21-s + (1.67e5 − 9.66e4i)23-s + (1.36e5 − 2.36e5i)25-s + (−4.13e5 − 3.33e5i)27-s + 9.14e5i·29-s + (−4.24e5 + 7.35e5i)31-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.956i)3-s + (1.12 − 0.651i)5-s + (−0.901 − 0.433i)7-s + (−0.828 + 0.560i)9-s + (1.30 + 0.752i)11-s − 1.86·13-s + (0.953 + 0.887i)15-s + (0.945 + 0.546i)17-s + (0.631 + 1.09i)19-s + (0.150 − 0.988i)21-s + (0.598 − 0.345i)23-s + (0.349 − 0.604i)25-s + (−0.778 − 0.627i)27-s + 1.29i·29-s + (−0.459 + 0.795i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.20770 + 1.65418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20770 + 1.65418i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-23.7 - 77.4i)T \) |
| 7 | \( 1 + (2.16e3 + 1.04e3i)T \) |
| good | 5 | \( 1 + (-705. + 407. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.90e4 - 1.10e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 5.33e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-7.89e4 - 4.56e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.22e4 - 1.42e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.67e5 + 9.66e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 9.14e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (4.24e5 - 7.35e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-5.88e5 - 1.01e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.19e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.18e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (8.76e5 - 5.05e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.38e6 + 1.95e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-3.56e6 - 2.05e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.89e6 - 3.27e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.76e7 - 3.05e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.14e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (7.42e6 - 1.28e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (3.51e7 + 6.08e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 7.08e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (2.46e6 - 1.42e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 6.00e7T + 7.83e15T^{2} \) |
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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
−12.91116583437875725050261792352, −12.06431019114193965652807885483, −10.10931772684107289449581233750, −9.837109265754047943398765648720, −8.943472028917841347210512084837, −7.18425514151775076379812404394, −5.69962236620683614429271401140, −4.58832383636444051253269668758, −3.17257658722362669460710109957, −1.51164693419881916580206533369,
0.58096814047728452183244156354, 2.25396756069289815010986472897, 3.11448403468774330486815741229, 5.58844808093258513993249186089, 6.53810452502301195768842499567, 7.42874487664760170148079005064, 9.288304130414965273315367669479, 9.662581482275979263063879929747, 11.48629261273406938839843178513, 12.37594480443688172096769561771
