L(s) = 1 | + (−1.69 − 2.92i)2-s + (−4.5 + 7.79i)3-s + (10.2 − 17.8i)4-s + (27.2 + 47.2i)5-s + 30.4·6-s − 177.·8-s + (−40.5 − 70.1i)9-s + (92.1 − 159. i)10-s + (−240. + 417. i)11-s + (92.5 + 160. i)12-s + 512.·13-s − 490.·15-s + (−28.8 − 49.8i)16-s + (−295. + 511. i)17-s + (−136. + 237. i)18-s + (−1.22e3 − 2.12e3i)19-s + ⋯ |
L(s) = 1 | + (−0.298 − 0.517i)2-s + (−0.288 + 0.499i)3-s + (0.321 − 0.556i)4-s + (0.487 + 0.844i)5-s + 0.345·6-s − 0.981·8-s + (−0.166 − 0.288i)9-s + (0.291 − 0.504i)10-s + (−0.600 + 1.03i)11-s + (0.185 + 0.321i)12-s + 0.841·13-s − 0.563·15-s + (−0.0281 − 0.0487i)16-s + (−0.247 + 0.429i)17-s + (−0.0995 + 0.172i)18-s + (−0.778 − 1.34i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4272756531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4272756531\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + (1.69 + 2.92i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-27.2 - 47.2i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (240. - 417. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 512.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (295. - 511. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.22e3 + 2.12e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (887. + 1.53e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 4.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (4.88e3 - 8.45e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-4.98e3 - 8.63e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 3.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (660. + 1.14e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.74e4 - 3.01e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (5.79e3 - 1.00e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.57e4 + 2.71e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.37e4 - 2.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (7.96e3 - 1.38e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.35e4 + 7.54e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 9.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-6.32e4 - 1.09e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.58e9T^{2} \) |
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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
−12.31277774357220334344641244209, −11.01536683377444152460517496901, −10.64372487067477541015769682322, −9.808737808358608613169708692594, −8.740011680516069725031613548174, −6.91571221903603225218304722775, −6.12375346337238107823753870727, −4.77190611721159090346440259307, −2.99475833392667575633572341474, −1.79744034849521019901075873248,
0.15158850471808925230315670640, 1.86605140714313862321342904308, 3.61012847975474235171968712627, 5.56038788476525639367237296016, 6.21472331689765563146594048875, 7.66633950519096638712929255769, 8.390587233359269993941059059446, 9.353872154033854311703440931671, 10.93190440561484011253255725951, 11.78851408930902049353863354741