L(s) = 1 | − 2.39·2-s + 3.73·4-s + 3.93·5-s + 7-s − 4.14·8-s − 9.42·10-s + 2.99·13-s − 2.39·14-s + 2.45·16-s − 6.60·17-s + 5.90·19-s + 14.6·20-s + 6.02·23-s + 10.5·25-s − 7.17·26-s + 3.73·28-s − 1.52·29-s + 8.46·31-s + 2.40·32-s + 15.8·34-s + 3.93·35-s − 0.607·37-s − 14.1·38-s − 16.3·40-s − 1.70·41-s + 3.23·43-s − 14.4·46-s + ⋯ |
L(s) = 1 | − 1.69·2-s + 1.86·4-s + 1.76·5-s + 0.377·7-s − 1.46·8-s − 2.98·10-s + 0.831·13-s − 0.639·14-s + 0.614·16-s − 1.60·17-s + 1.35·19-s + 3.28·20-s + 1.25·23-s + 2.10·25-s − 1.40·26-s + 0.705·28-s − 0.282·29-s + 1.52·31-s + 0.424·32-s + 2.71·34-s + 0.665·35-s − 0.0999·37-s − 2.29·38-s − 2.58·40-s − 0.266·41-s + 0.493·43-s − 2.12·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.602228686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.602228686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 + 0.607T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 + 0.599T + 67T^{2} \) |
| 71 | \( 1 + 1.40T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 - 2.55T + 97T^{2} \) |
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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
−8.129844760710982314508477911577, −7.16024661451463712324281853746, −6.71209171606926494206670693702, −6.04227105964025401467454392195, −5.32810343983897761353802382063, −4.45884951003760506790717199610, −2.96424391970384107784996441060, −2.33952851171055968153378125752, −1.50661950918317011526586963203, −0.904534672416759829732917227783,
0.904534672416759829732917227783, 1.50661950918317011526586963203, 2.33952851171055968153378125752, 2.96424391970384107784996441060, 4.45884951003760506790717199610, 5.32810343983897761353802382063, 6.04227105964025401467454392195, 6.71209171606926494206670693702, 7.16024661451463712324281853746, 8.129844760710982314508477911577