| L(s) = 1 | + (−0.342 + 0.939i)2-s + (0.0584 − 1.73i)3-s + (−0.766 − 0.642i)4-s + (−0.439 + 2.49i)5-s + (1.60 + 0.646i)6-s + (0.411 − 2.61i)7-s + (0.866 − 0.500i)8-s + (−2.99 − 0.202i)9-s + (−2.19 − 1.26i)10-s + (5.67 − 1.00i)11-s + (−1.15 + 1.28i)12-s + (−0.939 − 2.58i)13-s + (2.31 + 1.28i)14-s + (4.28 + 0.906i)15-s + (0.173 + 0.984i)16-s + (1.32 − 2.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (0.0337 − 0.999i)3-s + (−0.383 − 0.321i)4-s + (−0.196 + 1.11i)5-s + (0.655 + 0.264i)6-s + (0.155 − 0.987i)7-s + (0.306 − 0.176i)8-s + (−0.997 − 0.0674i)9-s + (−0.693 − 0.400i)10-s + (1.71 − 0.301i)11-s + (−0.334 + 0.371i)12-s + (−0.260 − 0.715i)13-s + (0.618 + 0.342i)14-s + (1.10 + 0.234i)15-s + (0.0434 + 0.246i)16-s + (0.320 − 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11381 - 0.313270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11381 - 0.313270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 3 | \( 1 + (-0.0584 + 1.73i)T \) |
| 7 | \( 1 + (-0.411 + 2.61i)T \) |
| good | 5 | \( 1 + (0.439 - 2.49i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-5.67 + 1.00i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.939 + 2.58i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.32 + 2.29i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0684 - 0.0395i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0439 + 0.0524i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.25 + 8.94i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-6.33 + 7.55i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.36 - 4.08i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.92 + 1.06i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.166 - 0.944i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.94 - 4.98i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 + (1.48 - 8.42i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.31 + 1.57i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (0.412 - 0.150i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.39 + 3.69i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.57 - 4.37i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.93 + 0.705i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.09 - 1.49i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-8.31 - 14.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.68 - 1.00i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.37815332544280480957583097081, −10.35721012397476659212783942784, −9.364728106490847286470313899677, −8.069875050607039527036532093167, −7.47219702379095233051936466129, −6.62310462775174033904720884785, −6.05504232408160498740438388215, −4.27030702373740102941158053143, −2.93810148645942692730068145126, −0.968472694167237133551123087728,
1.61170324443451594516637282995, 3.36046830105826253222213362033, 4.46454359907457573321090403826, 5.16142369760757327318067494574, 6.56994854635155305544233061274, 8.420130359087124581151691278985, 8.871718741222762696345535118052, 9.438606639767289394263081903738, 10.44517567636474008645574404691, 11.69345039281596206440778613324
