| L(s) = 1 | + (0.429 − 0.0913i)2-s + (0.335 − 1.69i)3-s + (−1.65 + 0.735i)4-s + (−1.76 − 1.36i)5-s + (−0.0109 − 0.760i)6-s + (−0.888 − 1.53i)7-s + (−1.35 + 0.982i)8-s + (−2.77 − 1.14i)9-s + (−0.885 − 0.425i)10-s + (4.21 − 0.895i)11-s + (0.694 + 3.05i)12-s + (−6.27 − 1.33i)13-s + (−0.522 − 0.579i)14-s + (−2.91 + 2.54i)15-s + (1.92 − 2.14i)16-s + (0.162 − 0.118i)17-s + ⋯ |
| L(s) = 1 | + (0.303 − 0.0645i)2-s + (0.193 − 0.981i)3-s + (−0.825 + 0.367i)4-s + (−0.791 − 0.611i)5-s + (−0.00446 − 0.310i)6-s + (−0.335 − 0.581i)7-s + (−0.478 + 0.347i)8-s + (−0.924 − 0.380i)9-s + (−0.279 − 0.134i)10-s + (1.26 − 0.269i)11-s + (0.200 + 0.881i)12-s + (−1.74 − 0.369i)13-s + (−0.139 − 0.154i)14-s + (−0.753 + 0.657i)15-s + (0.481 − 0.535i)16-s + (0.0395 − 0.0287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.286334 - 0.772267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.286334 - 0.772267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.335 + 1.69i)T \) |
| 5 | \( 1 + (1.76 + 1.36i)T \) |
| good | 2 | \( 1 + (-0.429 + 0.0913i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (0.888 + 1.53i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.21 + 0.895i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (6.27 + 1.33i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.162 + 0.118i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.79 + 3.48i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 2.03i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.0773 + 0.735i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (0.289 + 2.75i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (1.36 + 4.21i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-12.3 - 2.61i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (3.91 + 6.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0518 - 0.493i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-5.99 - 4.35i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (9.52 + 2.02i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (9.51 - 2.02i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.0149 + 0.141i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-1.95 - 1.41i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.860 - 2.64i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.34 - 12.7i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-3.13 - 1.39i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.29 + 10.1i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.65 + 15.7i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.14340421107266642983958845414, −11.42414143082954279569235176885, −9.532812416276354449331812656994, −8.931186802447378671827980162404, −7.68556126318465741316015865898, −7.15548360030558150758718569455, −5.47710736342166051376945518458, −4.25972730310862628328875707065, −3.09266965104391091818712969343, −0.63382424947647006242713611932,
3.03523582772307085497534197825, 4.14388478769693578212847889894, 5.02634211347520659503166412017, 6.34101877452598431385546924339, 7.71383149734483438350758336359, 9.088588987982817377872862066349, 9.562845726982192665503596507479, 10.51131399911384077869698335501, 11.86132489397014026806398753040, 12.31404372636336647288160682058
