L(s) = 1 | + (20.8 − 36.0i)5-s + (−101. − 176. i)7-s + (−235. − 407. i)11-s + (−241. + 418. i)13-s − 1.25e3·17-s − 1.97e3·19-s + (−239. + 414. i)23-s + (697. + 1.20e3i)25-s + (−580. − 1.00e3i)29-s + (−1.18e3 + 2.05e3i)31-s − 8.45e3·35-s + 8.18e3·37-s + (8.75e3 − 1.51e4i)41-s + (1.14e4 + 1.98e4i)43-s + (8.68e3 + 1.50e4i)47-s + ⋯ |
L(s) = 1 | + (0.372 − 0.644i)5-s + (−0.784 − 1.35i)7-s + (−0.585 − 1.01i)11-s + (−0.396 + 0.686i)13-s − 1.05·17-s − 1.25·19-s + (−0.0942 + 0.163i)23-s + (0.223 + 0.386i)25-s + (−0.128 − 0.221i)29-s + (−0.221 + 0.384i)31-s − 1.16·35-s + 0.982·37-s + (0.813 − 1.40i)41-s + (0.943 + 1.63i)43-s + (0.573 + 0.993i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0177 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2315593100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2315593100\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-20.8 + 36.0i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (101. + 176. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (235. + 407. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (241. - 418. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.97e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (239. - 414. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (580. + 1.00e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.18e3 - 2.05e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 8.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-8.75e3 + 1.51e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.14e4 - 1.98e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-8.68e3 - 1.50e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 5.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.22e4 + 3.85e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.08e3 + 3.60e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.22e3 + 2.12e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.03e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.52e4 + 4.37e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.59e4 - 4.48e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.02e4 + 6.96e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−10.69509940992428564682275816537, −9.588318109265337489915737894272, −8.910793657023083011274764723664, −7.80652741627335231697714918900, −6.79387454859138489963039332859, −5.96967187323461203002747921492, −4.65250037708897745202128830289, −3.82615732640285928815240145330, −2.42736747354408584243923560268, −0.930786149268317625249634598731,
0.06299316940170122638926302094, 2.36487502427848206386709121250, 2.56208549149007301377412313268, 4.26795831277447493427659176952, 5.51000754333390835869965852736, 6.31698761536369721772434594085, 7.18465863685527604614791969440, 8.416577094388365559323712522518, 9.277567304496774012119738583960, 10.13186355733009299860219491997