| L(s) = 1 | + (1.35 − 0.390i)2-s + (0.290 + 0.351i)3-s + (1.69 − 1.06i)4-s + (1.00 + 1.99i)5-s + (0.532 + 0.364i)6-s + (3.27 − 1.06i)7-s + (1.88 − 2.10i)8-s + (0.523 − 2.74i)9-s + (2.15 + 2.31i)10-s + (−0.505 + 4.00i)11-s + (0.866 + 0.287i)12-s + (−0.348 + 1.82i)13-s + (4.03 − 2.72i)14-s + (−0.407 + 0.935i)15-s + (1.74 − 3.59i)16-s + (−0.351 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (0.961 − 0.276i)2-s + (0.167 + 0.203i)3-s + (0.847 − 0.530i)4-s + (0.451 + 0.892i)5-s + (0.217 + 0.148i)6-s + (1.23 − 0.402i)7-s + (0.667 − 0.744i)8-s + (0.174 − 0.914i)9-s + (0.680 + 0.732i)10-s + (−0.152 + 1.20i)11-s + (0.250 + 0.0828i)12-s + (−0.0967 + 0.507i)13-s + (1.07 − 0.729i)14-s + (−0.105 + 0.241i)15-s + (0.436 − 0.899i)16-s + (−0.0851 − 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.69051 - 0.193365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.69051 - 0.193365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 + 0.390i)T \) |
| 5 | \( 1 + (-1.00 - 1.99i)T \) |
| good | 3 | \( 1 + (-0.290 - 0.351i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (-3.27 + 1.06i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.505 - 4.00i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (0.348 - 1.82i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (0.351 + 1.36i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (4.46 + 3.69i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (2.55 - 2.72i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (8.15 - 0.513i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (-5.40 + 1.38i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-4.18 - 2.30i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (2.71 - 2.55i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.804 - 0.584i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (2.83 + 7.16i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (0.588 - 0.928i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (-6.95 + 3.27i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (-4.68 + 4.98i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.294 - 4.68i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (0.689 - 0.272i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-10.0 - 4.71i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-1.98 - 2.39i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (7.58 - 9.17i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (5.18 - 11.0i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (15.5 - 0.979i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.920636861673252069338650157312, −9.619138393458836664334774445512, −8.119007330585495876855219412901, −7.02386482116754883050892941261, −6.70975753130965358560655998595, −5.47155423003557253181456071091, −4.51260159855486815928478559736, −3.86219499188602296694269930052, −2.52113917617896729264078295140, −1.67987101090665534995700565782,
1.63337242325594792491838685166, 2.45875845395848842211350674411, 3.98731969900334229284631838116, 4.87227655334763947664478227999, 5.57539097919316991024421843568, 6.20742932451619485162669783281, 7.71007628723657194494630345624, 8.215190690050056385854052176729, 8.678386259077304992298819466113, 10.24495555548416155219588972766
