| L(s) = 1 | + (1.37 + 4.23i)2-s + (2.00 + 1.45i)3-s + (9.83 − 7.14i)4-s + (7.72 − 23.7i)5-s + (−3.40 + 10.4i)6-s + (66.7 − 48.4i)7-s + (159. + 115. i)8-s + (−73.2 − 225. i)9-s + 111.·10-s + (315. − 247. i)11-s + 30.0·12-s + (150. + 463. i)13-s + (297. + 215. i)14-s + (50.0 − 36.3i)15-s + (−150. + 463. i)16-s + (−238. + 733. i)17-s + ⋯ |
| L(s) = 1 | + (0.243 + 0.748i)2-s + (0.128 + 0.0932i)3-s + (0.307 − 0.223i)4-s + (0.138 − 0.425i)5-s + (−0.0385 + 0.118i)6-s + (0.514 − 0.373i)7-s + (0.879 + 0.638i)8-s + (−0.301 − 0.927i)9-s + 0.352·10-s + (0.786 − 0.617i)11-s + 0.0602·12-s + (0.247 + 0.760i)13-s + (0.405 + 0.294i)14-s + (0.0573 − 0.0416i)15-s + (−0.146 + 0.452i)16-s + (−0.199 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.45609 + 0.378595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45609 + 0.378595i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-7.72 + 23.7i)T \) |
| 11 | \( 1 + (-315. + 247. i)T \) |
| good | 2 | \( 1 + (-1.37 - 4.23i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-2.00 - 1.45i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + (-66.7 + 48.4i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-150. - 463. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (238. - 733. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.28e3 - 930. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 373.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (2.08e3 - 1.51e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.60e3 + 4.95e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-3.38e3 + 2.45e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.10e4 + 8.00e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (7.98e3 + 5.80e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-6.13e3 - 1.88e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.59e4 - 2.61e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (50.2 - 154. i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 5.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-96.4 + 296. i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.21e4 + 2.33e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.71e4 - 5.26e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.61e4 - 8.04e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.55e4 + 7.87e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.42423712493962334589903092217, −13.70747825722867888886364345708, −11.98701018286024104132345705228, −11.01978920807029658421187404297, −9.429137374901667806730011623419, −8.188309791086785193792445817893, −6.72502546137731129769680020298, −5.65809584886782120371428196255, −3.98477275461649259879589679472, −1.41512828704061948525958608883,
1.79681151699732296423565521868, 3.13046442763220357234694584423, 4.98010777882426421312085552237, 6.91361421596102490033196095152, 8.114010024423638262790593367700, 9.816581735721740331520105885184, 11.07560016972340344620110201089, 11.75461120481866762278176173126, 13.04102070601211681200796291752, 14.03240788861417294999296321730
