| L(s) = 1 | + 2-s − 2.35·3-s + 4-s − 2.03·5-s − 2.35·6-s − 2.66·7-s + 8-s + 2.54·9-s − 2.03·10-s + 11-s − 2.35·12-s − 2.32·13-s − 2.66·14-s + 4.79·15-s + 16-s + 5.61·17-s + 2.54·18-s + 5.93·19-s − 2.03·20-s + 6.26·21-s + 22-s − 7.13·23-s − 2.35·24-s − 0.851·25-s − 2.32·26-s + 1.07·27-s − 2.66·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.35·3-s + 0.5·4-s − 0.910·5-s − 0.961·6-s − 1.00·7-s + 0.353·8-s + 0.847·9-s − 0.644·10-s + 0.301·11-s − 0.679·12-s − 0.643·13-s − 0.710·14-s + 1.23·15-s + 0.250·16-s + 1.36·17-s + 0.599·18-s + 1.36·19-s − 0.455·20-s + 1.36·21-s + 0.213·22-s − 1.48·23-s − 0.480·24-s − 0.170·25-s − 0.455·26-s + 0.207·27-s − 0.502·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059724964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059724964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.66T + 7T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 4.68T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 - 6.98T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 47 | \( 1 + 0.808T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 - 9.80T + 61T^{2} \) |
| 67 | \( 1 + 4.04T + 67T^{2} \) |
| 71 | \( 1 - 0.814T + 71T^{2} \) |
| 73 | \( 1 - 7.75T + 73T^{2} \) |
| 79 | \( 1 + 7.57T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 - 6.34T + 97T^{2} \) |
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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
−10.07723023237188297663866838334, −9.669054504875722392490639083148, −7.973809308186264975142959595729, −7.35645789081847855151233936682, −6.29809639203341677285747226693, −5.81422740105032919552879406789, −4.81195360799258178450332184997, −3.89594481694876336927647391784, −2.91082318584403514033682976451, −0.77431551016974618164193974756,
0.77431551016974618164193974756, 2.91082318584403514033682976451, 3.89594481694876336927647391784, 4.81195360799258178450332184997, 5.81422740105032919552879406789, 6.29809639203341677285747226693, 7.35645789081847855151233936682, 7.973809308186264975142959595729, 9.669054504875722392490639083148, 10.07723023237188297663866838334
