| L(s) = 1 | + (1.17 + 0.789i)2-s + (−1.69 + 0.357i)3-s + (0.752 + 1.85i)4-s + (1.62 − 2.46i)5-s + (−2.27 − 0.919i)6-s + (1.06 − 1.00i)7-s + (−0.579 + 2.76i)8-s + (2.74 − 1.21i)9-s + (3.85 − 1.61i)10-s + (4.08 + 2.04i)11-s + (−1.93 − 2.87i)12-s + (0.0903 − 0.209i)13-s + (2.03 − 0.336i)14-s + (−1.87 + 4.76i)15-s + (−2.86 + 2.79i)16-s + (−3.16 + 3.77i)17-s + ⋯ |
| L(s) = 1 | + (0.829 + 0.558i)2-s + (−0.978 + 0.206i)3-s + (0.376 + 0.926i)4-s + (0.726 − 1.10i)5-s + (−0.926 − 0.375i)6-s + (0.401 − 0.378i)7-s + (−0.204 + 0.978i)8-s + (0.914 − 0.403i)9-s + (1.21 − 0.510i)10-s + (1.23 + 0.617i)11-s + (−0.559 − 0.828i)12-s + (0.0250 − 0.0580i)13-s + (0.544 − 0.0900i)14-s + (−0.482 + 1.23i)15-s + (−0.716 + 0.697i)16-s + (−0.767 + 0.915i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73481 + 0.645529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73481 + 0.645529i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 - 0.789i)T \) |
| 3 | \( 1 + (1.69 - 0.357i)T \) |
| good | 5 | \( 1 + (-1.62 + 2.46i)T + (-1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-1.06 + 1.00i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-4.08 - 2.04i)T + (6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-0.0903 + 0.209i)T + (-8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (3.16 - 3.77i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (0.207 + 0.247i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.349 - 0.370i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.447 + 3.83i)T + (-28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-2.18 + 9.21i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (0.158 - 0.900i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-0.900 + 0.670i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-8.72 + 0.507i)T + (42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (12.3 - 2.92i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (4.09 + 2.36i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.59 - 2.81i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-3.39 + 11.3i)T + (-50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-0.746 - 6.38i)T + (-65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (14.0 - 5.13i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (11.6 + 4.23i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (7.56 + 5.63i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-2.24 + 3.01i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (1.73 - 4.76i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.90 + 1.24i)T + (38.4 - 89.0i)T^{2} \) |
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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
−11.84690972573532856235028978483, −11.14001951511210082159077448984, −9.824069949180694039695215870049, −8.918458162960130081274296233277, −7.67525081671686263377096740843, −6.44342402903750681733911825568, −5.84549006242577975352213229435, −4.60703981672851866392311764097, −4.19134549163585074612368758813, −1.66069937007043402520600491669,
1.59072304488020518151001015537, 2.99874580207959894412605234140, 4.50928429020220094977905589922, 5.60554859764320008176858358881, 6.47904422995509266761933320662, 6.99972855334112602182250024241, 9.013567265671575892010206984228, 10.09275270862163373809852944813, 10.89679300120421603465384441404, 11.48829090726094625698751859946
