L(s) = 1 | − 2-s − 2.38·3-s + 4-s + 2.54·5-s + 2.38·6-s + 3.05·7-s − 8-s + 2.69·9-s − 2.54·10-s + 5.40·11-s − 2.38·12-s + 2.16·13-s − 3.05·14-s − 6.06·15-s + 16-s + 0.860·17-s − 2.69·18-s − 6.33·19-s + 2.54·20-s − 7.28·21-s − 5.40·22-s − 2.60·23-s + 2.38·24-s + 1.46·25-s − 2.16·26-s + 0.739·27-s + 3.05·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.37·3-s + 0.5·4-s + 1.13·5-s + 0.973·6-s + 1.15·7-s − 0.353·8-s + 0.896·9-s − 0.803·10-s + 1.63·11-s − 0.688·12-s + 0.599·13-s − 0.816·14-s − 1.56·15-s + 0.250·16-s + 0.208·17-s − 0.634·18-s − 1.45·19-s + 0.568·20-s − 1.59·21-s − 1.15·22-s − 0.542·23-s + 0.486·24-s + 0.292·25-s − 0.423·26-s + 0.142·27-s + 0.577·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.403258798\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.403258798\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 4021 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 5.40T + 11T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 - 0.860T + 17T^{2} \) |
| 19 | \( 1 + 6.33T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 9.11T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 + 8.15T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.87433212832950000250539549473, −6.85446417126600161510516220524, −6.40699663486765031473163620466, −5.91130045429054122161840123091, −5.28558923925155340290156807155, −4.46109284197093333483090408207, −3.65424137583157741645575134080, −2.00905119263080677945380198717, −1.68806621939473063487071328002, −0.73930823400713310815092294811,
0.73930823400713310815092294811, 1.68806621939473063487071328002, 2.00905119263080677945380198717, 3.65424137583157741645575134080, 4.46109284197093333483090408207, 5.28558923925155340290156807155, 5.91130045429054122161840123091, 6.40699663486765031473163620466, 6.85446417126600161510516220524, 7.87433212832950000250539549473