| L(s) = 1 | + (3.07 + 80.9i)3-s + (304. − 175. i)5-s + (517. + 2.34e3i)7-s + (−6.54e3 + 497. i)9-s + (−1.89e4 − 1.09e4i)11-s + 5.25e3·13-s + (1.51e4 + 2.40e4i)15-s + (−5.36e3 − 3.09e3i)17-s + (5.18e4 + 8.97e4i)19-s + (−1.88e5 + 4.90e4i)21-s + (−2.37e5 + 1.37e5i)23-s + (−1.33e5 + 2.31e5i)25-s + (−6.03e4 − 5.28e5i)27-s − 2.72e5i·29-s + (−5.39e4 + 9.33e4i)31-s + ⋯ |
| L(s) = 1 | + (0.0379 + 0.999i)3-s + (0.486 − 0.280i)5-s + (0.215 + 0.976i)7-s + (−0.997 + 0.0758i)9-s + (−1.29 − 0.747i)11-s + 0.184·13-s + (0.299 + 0.475i)15-s + (−0.0642 − 0.0370i)17-s + (0.397 + 0.688i)19-s + (−0.967 + 0.252i)21-s + (−0.849 + 0.490i)23-s + (−0.342 + 0.592i)25-s + (−0.113 − 0.993i)27-s − 0.384i·29-s + (−0.0583 + 0.101i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.104387 - 0.279263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104387 - 0.279263i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.07 - 80.9i)T \) |
| 7 | \( 1 + (-517. - 2.34e3i)T \) |
| good | 5 | \( 1 + (-304. + 175. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.89e4 + 1.09e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 5.25e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + (5.36e3 + 3.09e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-5.18e4 - 8.97e4i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.37e5 - 1.37e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 2.72e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (5.39e4 - 9.33e4i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.46e6 + 2.54e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.88e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.82e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.87e6 + 2.23e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (8.81e6 + 5.08e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-3.48e6 - 2.01e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.96e6 + 8.59e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.65e6 - 2.86e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.30e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.72e7 - 2.98e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.99e6 + 5.18e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 6.41e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-5.01e7 + 2.89e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 6.99e7T + 7.83e15T^{2} \) |
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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
−13.42284405032717183853043028476, −12.09146244614803098234931377446, −10.99044626273330655147959204037, −9.965740834179333015098495876550, −8.944033928690357311373720502988, −7.996574305217543617599964824039, −5.75461877873345778379442010223, −5.28529038361305033646974071010, −3.53818768616521436449972427850, −2.16085529989449915148206587123,
0.081932846343111987391978374751, 1.59204468934563196990791914112, 2.85816873677089025143989738922, 4.81319784933584992516118993740, 6.29879089182056969037849660383, 7.34855535678550532305838622255, 8.231398423730500672110503960756, 9.915574434857278200086342865942, 10.84111801817737469825393516625, 12.11399825662440851126632789587
