L(s) = 1 | − 2-s + (1.68 − 0.420i)3-s + 4-s + (1.08 + 1.95i)5-s + (−1.68 + 0.420i)6-s + (−0.595 + 2.57i)7-s − 8-s + (2.64 − 1.41i)9-s + (−1.08 − 1.95i)10-s + 2.82i·11-s + (1.68 − 0.420i)12-s − 3.36·13-s + (0.595 − 2.57i)14-s + (2.64 + 2.82i)15-s + 16-s − 4.75i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.970 − 0.242i)3-s + 0.5·4-s + (0.485 + 0.874i)5-s + (−0.685 + 0.171i)6-s + (−0.224 + 0.974i)7-s − 0.353·8-s + (0.881 − 0.471i)9-s + (−0.343 − 0.618i)10-s + 0.852i·11-s + (0.485 − 0.121i)12-s − 0.931·13-s + (0.159 − 0.688i)14-s + (0.683 + 0.730i)15-s + 0.250·16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19825 + 0.322059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19825 + 0.322059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.68 + 0.420i)T \) |
| 5 | \( 1 + (-1.08 - 1.95i)T \) |
| 7 | \( 1 + (0.595 - 2.57i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 + 0.500iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 + 3.32iT - 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 7.82iT - 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 9.81iT - 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.97iT - 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−12.44018196176453478263653381322, −11.43027773455827526448793429514, −10.12301414630759974824149943883, −9.418941661935215518324954870148, −8.746704108898883758402176480287, −7.13145911071710801752328188035, −6.98412510226340640997351694651, −5.13951110018295177027555838325, −2.95887419098214329582836339777, −2.24690450593217753132228586265,
1.47576266495894743182816823569, 3.26538884246364051243265404772, 4.65619394929751210181300840883, 6.26150910877997846109630329643, 7.66361857873678680843488859850, 8.345812091449368749717633196133, 9.377463996036261287268372059865, 10.04648846558891200247002691683, 10.94446956097126160803550353724, 12.52964102425053588922341719864