L(s) = 1 | + (−2.72 − 4.71i)2-s + (−4.5 + 7.79i)3-s + (1.16 − 2.02i)4-s + (18 + 31.1i)5-s + 49.0·6-s − 187.·8-s + (−40.5 − 70.1i)9-s + (98.0 − 169. i)10-s + (−92.2 + 159. i)11-s + (10.5 + 18.2i)12-s − 147.·13-s − 324·15-s + (471. + 817. i)16-s + (984. − 1.70e3i)17-s + (−220. + 382. i)18-s + (946. + 1.63e3i)19-s + ⋯ |
L(s) = 1 | + (−0.481 − 0.833i)2-s + (−0.288 + 0.499i)3-s + (0.0365 − 0.0632i)4-s + (0.321 + 0.557i)5-s + 0.555·6-s − 1.03·8-s + (−0.166 − 0.288i)9-s + (0.310 − 0.536i)10-s + (−0.229 + 0.397i)11-s + (0.0210 + 0.0365i)12-s − 0.242·13-s − 0.371·15-s + (0.460 + 0.798i)16-s + (0.825 − 1.43i)17-s + (−0.160 + 0.277i)18-s + (0.601 + 1.04i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.074914322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074914322\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.72 + 4.71i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-18 - 31.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (92.2 - 159. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-984. + 1.70e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-946. - 1.63e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (68.4 + 118. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 1.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-4.48e3 + 7.76e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (6.44e3 + 1.11e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 8.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.00e4 + 1.73e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (4.66e3 - 8.08e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-4.43e3 + 7.67e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.05e4 + 3.56e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.76e4 + 4.79e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.06e4 + 3.57e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (8.48e3 + 1.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.35e4 - 7.54e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
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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
−11.68372654581346772852129316448, −10.62855220372190781064662122991, −9.944306621427415484763492560506, −9.284705836750548717170758606941, −7.65275987164542417806405588247, −6.26971477684641125710902287419, −5.16396460015153211399873043696, −3.38238524204712429410322952491, −2.18324051894513451154729981862, −0.49244682847355574768995362604,
1.19366849530245403822619531518, 3.08689152358515393756691772641, 5.13393218386062056853210059318, 6.15056555912752834632810224665, 7.18036492138922651352710120275, 8.206762316558817103292743599757, 8.979475802117528928242096572000, 10.28318524510038007904870296454, 11.62303347943035976234938067510, 12.50269142519634616817399501587