| L(s) = 1 | + (1.31 − 0.513i)2-s + (0.281 + 0.959i)3-s + (1.47 − 1.35i)4-s + (0.897 − 1.39i)5-s + (0.864 + 1.11i)6-s + (−0.549 − 3.81i)7-s + (1.24 − 2.54i)8-s + (−0.841 + 0.540i)9-s + (0.464 − 2.30i)10-s + (−2.54 + 5.58i)11-s + (1.71 + 1.03i)12-s + (−0.319 + 2.22i)13-s + (−2.68 − 4.75i)14-s + (1.59 + 0.467i)15-s + (0.333 − 3.98i)16-s + (3.31 − 2.87i)17-s + ⋯ |
| L(s) = 1 | + (0.931 − 0.363i)2-s + (0.162 + 0.553i)3-s + (0.735 − 0.677i)4-s + (0.401 − 0.624i)5-s + (0.352 + 0.457i)6-s + (−0.207 − 1.44i)7-s + (0.439 − 0.898i)8-s + (−0.280 + 0.180i)9-s + (0.147 − 0.727i)10-s + (−0.768 + 1.68i)11-s + (0.494 + 0.297i)12-s + (−0.0886 + 0.616i)13-s + (−0.717 − 1.26i)14-s + (0.411 + 0.120i)15-s + (0.0833 − 0.996i)16-s + (0.805 − 0.697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.14304 - 0.748733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14304 - 0.748733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.31 + 0.513i)T \) |
| 3 | \( 1 + (-0.281 - 0.959i)T \) |
| 23 | \( 1 + (-4.03 - 2.59i)T \) |
| good | 5 | \( 1 + (-0.897 + 1.39i)T + (-2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.549 + 3.81i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.54 - 5.58i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.319 - 2.22i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.31 + 2.87i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (4.30 - 4.96i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.25 - 3.75i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.0287 - 0.0979i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (4.07 + 6.34i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (0.593 + 0.381i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (8.96 - 2.63i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 4.98iT - 47T^{2} \) |
| 53 | \( 1 + (6.22 - 0.894i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-8.20 - 1.17i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.380 + 1.29i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (5.11 + 11.2i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (3.79 - 1.73i)T + (46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.75 - 3.18i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.84 + 12.8i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 8.71i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.77 - 9.45i)T + (-74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-3.86 + 6.01i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−11.99252093886130283074809079354, −10.63146048657429756822237248445, −10.15832451919568544766541478539, −9.350918486632335911753498275440, −7.61170540515122933344689140479, −6.80490578107940729260553244413, −5.20851345280016116866464971380, −4.57485307269535046075160154383, −3.48126513859295197910636982062, −1.73474305824984758210183807298,
2.56877532455535116005292963789, 3.09780795254805466248888777598, 5.17984195651400356248422543247, 6.02604079620050467059905159725, 6.64810326477345428146531935371, 8.205492952753539116942465373628, 8.607952208337133960398802929003, 10.39926335359800418500548159898, 11.26397459693346331140455525858, 12.23040629739941168290009447345
