L(s) = 1 | + (−0.203 + 1.39i)2-s − 1.12i·3-s + (−1.91 − 0.568i)4-s − 2.23·5-s + (1.57 + 0.227i)6-s + (1.18 − 2.56i)8-s + 1.74·9-s + (0.453 − 3.12i)10-s + 5.95i·11-s + (−0.637 + 2.15i)12-s + 6.51·13-s + 2.50i·15-s + (3.35 + 2.17i)16-s + (−0.353 + 2.43i)18-s + 4.35i·19-s + (4.28 + 1.27i)20-s + ⋯ |
L(s) = 1 | + (−0.143 + 0.989i)2-s − 0.647i·3-s + (−0.958 − 0.284i)4-s − 0.999·5-s + (0.641 + 0.0930i)6-s + (0.418 − 0.908i)8-s + 0.580·9-s + (0.143 − 0.989i)10-s + 1.79i·11-s + (−0.184 + 0.621i)12-s + 1.80·13-s + 0.647i·15-s + (0.838 + 0.544i)16-s + (−0.0832 + 0.574i)18-s + 0.999i·19-s + (0.958 + 0.284i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.836269 + 0.624385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.836269 + 0.624385i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.203 - 1.39i)T \) |
| 5 | \( 1 + 2.23T \) |
| 19 | \( 1 - 4.35iT \) |
good | 3 | \( 1 + 1.12iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.95iT - 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.01T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 1.11T + 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−11.70965988934871158569501687148, −10.47062014236348868872866181926, −9.552662669023053945827538661577, −8.367191465049300895786259112127, −7.71289619431099041117017866738, −6.96982503180036210989343269237, −6.11486326529476646654984497767, −4.60562065818393571572910093184, −3.83884039823818018222714491154, −1.38777519347960803787656745516,
0.923751885223943354440011888904, 3.20927807345127265803627934958, 3.80780112196605458523585040753, 4.84351286270871015910408530282, 6.24957611882866798165079138030, 7.85945440204638616762641532278, 8.631813473856212505258583269451, 9.310558072298132654524540175916, 10.67565768786094845398836671144, 11.05834075811814045706146179729